families of quadratics - glencoe.com

Investigation

1page T84

Investigation

2page T87

Investigation

3page T90

Investigation

4page T93

LabInvestigation

page T96

LessonPlanner

ObjectivesTo understand theeffects of a, b, and con the graphs ofparabolas of the formy � ax2 � bx � c

To use quadraticequations and graphsto analyze the motionof projectiles

To distinguish betweenquadratic relationshipsand other types ofrelationships, such ascubics

Overview (pacing: about 5 class periods)In this lesson, students explore the connection among three representations ofquadratics relationships: equations, tables, and graphs. They begin by looking attables of values for simple quadratics and examining how the graphs are affected bychanges in the equations. Then, after a brief introduction to the general form of aquadratic equation, y � ax2 � bx � c, students use graphing calculators to examinethe effects on the graphs of changing a, b, and c. They end the lesson with applicationsof quadratics and comparisons of quadratics to cubics and other types of equations.

Advance PreparationYou will need graphing calculators in Investigations 2–4 and the Lab Investigation.Master 43, graphing calculator grids, will be helpful for students to sketch their results.

Families of Quadratics

Summary

Students move between tables of data, equations,and graphs, seeing how changing coefficients in qua-dratic equations of the form y � ax2 � c changethe patterns in tables and graphs.

Students further explore the impact of changing a andc in quadratic equations of the form y � ax2 � c;then they explore the impact of changing b in quadraticequations of the form y � ax2 � bx � c.

Students use quadratic equations and graphs toanswer questions about the paths of thrown objects.

Students explore differences between linear, quadratic,and cubic equations.

Students manipulate coefficients of quadratic equationsto generate designs on a graphing calculator.

Assessment Opportunities

Share & Summarize, pages T86–T87,86–87

Troubleshooting, page T87

Share & Summarize, pages T89, 89Troubleshooting, page A364

On the Spot Assessment, page T91Share & Summarize, pages T92, 92Troubleshooting, page T92

Share & Summarize, pages T95, 95Troubleshooting, page T95Informal Assessment, page 106Quick Quiz, page 106

Materials

• Graphing calculators• Master 43• Length of wire (optional)

• Graphing calculators

• Graphing calculators

• Graphing calculators• Master 43

On Your Own Exercises

Practice & Apply: 1–7, pp. 98–99Connect & Extend: 26, p. 103Mixed Review: 32–44,

pp. 106–107

Practice & Apply: 8–12, pp. 99–100

Connect & Extend: 27, 28, pp. 103–104

Mixed Review: 32–44, pp. 106–107

Practice & Apply: 13, p. 101Connect & Extend: 29,

pp. 104–105Mixed Review: 32–44,

pp. 106–107

Practice & Apply: 14–25, p. 102Connect & Extend: 30, 31,

pp. 105–106Mixed Review: 32–44,

pp. 106–107

L E S S O N 2 . 2 Families of Quadratics 82a

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T83 C H A P T E R 2 Quadratic and Inverse Relationships

In this lesson, students work with quadratic equations inthe form y � ax2 � bx � c. This form of a quadraticequation can be quite confusing because letters areused to represent constants (a, b, and c) as well asvariables (x and y). This can be especially troublesomein the investigations when students begin to vary a, b,and c yet they are still referred to as constants.

It’s best to sort out the vocabulary early to avoidmisconceptions and confusion. You can use this example,writing it on the board:

2x2 � 1

Point out that this is a quadratic expression, not aquadratic equation, since it has no equals sign. The firstterm, 2x2, is called the x2 term, and 2 is the coefficientof the x2 term. You may want to offer other examplesof the form ax2 � c and ask students to identify thecoefficients of the x2 term. Whenever you can, refer tothe quantities in expressions or equations by name: thevariable x, the coefficient 2, the constant 1.

Explain that any quadratic expression can be written inthe form ax2 � bx � c, where a is not 0. Ask:

Why must a not be 0? If it were, there wouldbe no x2 term, and you would have a linearexpression instead.

In the expression 2x2 � 1, what is a? 2 b? 0 c? 1

Similarly, a quadratic equation can be written in the formy � ax2 � bx � c. In the previous lesson, studentslooked at some of these, such as y � 4.5x2, T � S2 � 1,and d � �

12�n2 � �

32�n. For each of these equations, ask

students to state the values of a, b, and c. If they havetrouble identifying the value of a in S2 � 1, you mightask what number S2 can be multiplied by to get S2.Then explain that all quadratic equations have symmet-ric, U-shaped graphs called parabolas.

Explain to students that they need to understand how toidentify and name a, b, and c since they play an impor-tant role in exploring the connection among graphs,tables, and equations for quadratic relationships—andthat is the focus of this lesson.

Introduce

➧1

➧2

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L E S S O N 2 . 2 Families of Quadratics 83

In Lesson 2.1, you explored several quadratic relationships:

y � x2 d � �n2 �

23n

� V � 4.5x2 T � S2 � 1

Any expression that can be written in the form ax2 � bx � c, where a, b,and c are constants, and a � 0, is called a quadratic expression in x.

The first term, ax2, is called the x2 term. The number multiplying x2, rep-resented by a, is called the coefficient of the x2 term. In the same way, bxis the x term and b is its coefficient. The coefficients a and b, and the con-stant c, can stand for positive or negative numbers, but a cannot be 0.(Can you see why?)

A quadratic equation can be written in the form y � ax2 � bx � c,where a � 0. Like those in Lesson 2.1, all quadratic equations have sym-metric, U-shaped graphs called parabolas.

Before you explore the connection among graphs, tables, and equationsfor quadratic relationships, it will help to review graphs, tables, and equa-tions for linear relationships.

Families ofQuadratics

V O C A B U L A R Yquadratic

expression

V O C A B U L A R Yquadratic

equation

Mirrors in the shape ofparabolas are used inflashlights and head-lights to focus lightinto a narrow beam.

factsJust

t h e


➧1 Talk about expressionsversus equations.

➧2 • Talk about why acan’t be 0.

• Ask students toidentify a, b, and cin specific quadraticequations.

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Think & DiscussMany students look at each new topic in mathematics asa completely new beginning. The more they can seeconnections and similarities with what they have previ-ously learned, the more they will see common threads inbigger mathematical ideas. This Think & Discuss helpsstudents to connect past work with linear equations andgraphs to what they are about to explore in this lesson,specifically, how different values for a and c in the equa-tion y � ax2 � c affect the position and width of thegraph of a quadratic relationship.

You may want to give students a few minutes to matchthe equations with their graphs and tables before goingover the questions as a class. This is not only a goodreview of their work with linear equations but also apreview of the analysis they will do with quadratics.

After discussing the questions, point out that the generalform of a linear equation is typically written y � mx � bbut could easily be written y � ax � b—in fact, students’graphing calculators might use a instead of m for the lineof best fit—to make the connection to the general formof a quadratic equation, y � ax2 � bx � c. You maywant to ask what the coefficient of x is in y � ax � b.

Investigation 1In this investigation, students use a table of values fory � x2 as a basis of comparison for tables representingother equations in the same family: y � x2 � 1,y � x2 � 1, and so on. They then use the tables tomatch each equation with the appropriate graph so thatthey begin to see connections among the equations,tables, and graphs.

Problem Set A Grouping: PairsIn this beginning activity, students use Column A (y � x2)as a basis for completing the columns for other equationsin the same family.

As students work on Problems 1 and 2, encouragethem to keep looking for connections to Column A asthey fill in the other values. For example, they may seethat to find a value for Column E corresponding to aparticular x value, they need to look for the value inColumn A that corresponds to the x value 1 greater. Forinstance, to find the value in Column E for x � 2.2, findthe value in Column A for x � 3.2. Similarly, to find avalue in Column F corresponding to a particular x value,find the value in Column A that corresponds to the xvalue 1 less.


Develop

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Investigation

② If m is positive, theline slopes up fromleft to right. If m isnegative, the lineslopes down fromleft to right. If m is 0,the line is horizontal.The greater the absolute value of m,the steeper thegraph.

&

84 C H A P T E R 2 Quadratic and Inverse Relationships

Think Discuss

Equations, graphs, and tables of four linear relationships are shownhere. Match each equation with itsgraph and table, and explain howyou made each match.

Equation 1: y � x

Equation 2: y � 2x

Equation 3: y � 2x � 3

Equation 4: y � �2x � 3

Table 1 Table 2

x �2 �1 0 1 2 3 x �2 �1 0 1 2 3

y 7 5 3 1 �1 �3 y �4 �2 0 2 4 6

Table 3 Table 4

x �2 �1 0 1 2 3 x �2 �1 0 1 2 3

y �2 �1 0 1 2 3 y �1 1 3 5 7 9

For linear equations of the form y � mx � b, how does the value ofm affect the graph?

For linear equations of the form y � mx � b, how does the value ofb affect the graph?

1 Quadratic Equations,Tables, and Graphs

Now you will explore the connection among equations, tables, and graphsfor quadratic relationships.

Problem Set A

1. The table on the next page has columns for six quadratic equations.Copy and complete the table to give the y values for the given x val-ues. (Hint: You can save time by completing Column A first, andthen looking for connections between its equation, y � x2, and theother equations. For example, if you know the value of x2 inColumn A, how can you easily determine x2 � 1 in Column B?)

The value of b is the point where the graphcrosses the y-axis. The line y � mx � bpasses through the y-axis at the point (0, b).

See ②.

See ①.

5

5

�5

�5

y

x

Graph 1

Graph 4

Graph 2

Graph 3

① Possible answers:Equation 1, Graph 2,Table 3; Graph 2matches Equation 1since it goes through theorigin and has slope 1.Table 3 matchesEquation 1 since the xand y values are equal.Equation 2, Graph 4,Table 2; Graph 4matches Equation 2since it goes through theorigin and has slope 2.Table 2 matchesEquation 2 since eachy value is twice the cor-responding x value.Equation 3, Graph 3,Table 4; Graph 3matches Equation 3since it has y-intercept 3and positive slope. Table4 matches Equation 3since it contains thepoint (0, 3) and the difference in consecutivey values is 2.Equation 4, Graph 1,Table 1; Graph 1matches Equation 4since it has y-intercept 3and negative slope.Table 1 matches Equation 4 since thedifference in consecutive y values is �2.


➧1 Discuss the questionsafter students havehad time to matchthe equations andgraphs.

➧2 Point out that thisequation can also bewritten y = ax + b.

➧3 • Have students workin pairs.

• Encourage studentsto look for connec-tions to Column Aas they fill in thetable.

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f o r a l l LearnersAccess

Extra Challenge The table is a wonderful medium for dis-covering other patterns and comparisons. For example, stu-dents may notice that the numbers in Columns E and F areexactly the same but in reverse order. The reason is some-what complex, and students may find it a challenge toexplain why.

In the table, x varies from �4 to 4, with all negative valuesmatched by positive values. As x varies from �4 to 4, x � 1varies from �5 to 3, and x � 1 varies from �3 to 5. Theresulting values are then squared, and the square of �3,the first number in Column E, is equal to that of 3, the lastnumber in Column F. Similarly, the squares of 5 and �5are the same, so the same numbers occur but in oppositepositions.

See the notes for Problem 2 on page T84.

In Problems 3–8, matching the tables to the graphsshould not be difficult, especially if students focus onclues such as high and low points. Encourage studentsto look for general patterns in the numbers. For example,all of the numbers in Column D are negative (studentsshould be able to explain why from the equation), andthe only graph that depicts this is that in Problem 8.

Problem Set Wrap-Up Students may find it helpful tohear each other’s problem-solving strategies for solvingthe problems in this set. Bring the class together andencourage a few students to briefly explain their ideas.


Develop

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A B C D E F

x y � x2 y � x2 � 1 y � x2 � 1 y � �x2 y � (x � 1)2 y � (x � 1)2

�4 16 17 15 �16 9 25�3.2 10.24 11.24 9.24 �10.24 4.84 17.64�2.2 4.84 5.84 3.84 �4.84 1.44 10.24

�1 1 2 0 �1 0 4�0.5 0.25 1.25 �0.75 �0.25 0.25 2.25

0 0 1 �1 0 1 10.5 0.25 1.25 �0.75 �0.25 2.25 0.251 1 2 0 �1 4 0

2.2 4.84 5.84 3.84 �4.84 10.24 1.443.2 10.24 11.24 9.24 �10.24 17.64 4.844 16 17 15 �16 25 9

2. Compare the values in each of Columns B–F with those in Column A.Explain how each comparison you make is related to the equations.

For the graphs in Problems 3–8, complete Parts a–c.

3. 4. 5.

6. 7. 8.

a. Match each graph with one of the quadratic equations in ColumnsA–F, and explain your reasoning.

b. Describe how the graph differs from the graph of y � x2.

c. Describe the graph’s line of symmetry.

y

x

y

x

y

x

y

x

y

x

y

x

3–4. See below.

RememberThe expression �n2

means �(n2), not(�n)2. To calculate �n2,square n and take theopposite of the result.

2. The values inColumn B are 1more than those inColumn A, since theequation for ColumnB is y � x2 � 1 andthat of Column A isy � x2.The values inColumn C are 1 lessthan those inColumn A, since theequation for ColumnC is y � x2 � 1.The values inColumn D are thenegatives of thosein Column A, sincethe equation forColumn D is y ��x2. The values inColumn E are thesquares of 1 morethan the numbersthat were squaredfor Column A, sincethe equation for

Column E isy � (x � 1)2.The values inColumn F are thesquares of 1 lessthan the numbersthat were squaredfor Column A,since the equationfor Column F isy � (x � 1)2.

5–8. See Additional Answers.

3. a. Column A, y � x2, since the lowest value is at (0, 0).b. It is the graph of y � x2.c. the y-axis

4. a. Column E, y � (x � 1)2, since the lowest y value is 0, when x � �1.b. It has the same shape as y � x2 but is shifted to the left.c. the vertical line passing through (�1, 0)


Additional Answers5. a. Column C, y � x2 � 1, since the lowest y value is �1, when

x � 0.b. It has the same shape as y � x2 but is shifted down.c. the y-axis

6. a. Column B, y � x2 � 1, since the lowest y value is 1, when x � 0.b. It has the same shape as y � x2 but is shifted up.c. the y-axis

7. a. Column F, y � (x � 1)2, since the lowest y value is 0, whenx � 1.

b. It has the same shape as y � x2 but is shifted to the right.c. the vertical line passing through (1, 0)

8. a. Column D, y � �x2, since the greatest y value is 0, when x � 0,and all the other y values are negative.

b. It has the same shape as y � x2 but is flipped upside down.c. the y-axis

➧1 Problem-Solving StrategyLook for a pattern

➧2 Have students sharetheir problem-solvingstrategies for ProblemSet A.

082a-107a TE 02/2 MSMgr8 5/22/04 9:30 AM


Problem Set B Suggested Grouping: PairsThis problem set challenges students to find equations forquadratic relationships by comparing tables of data.

• Students should easily recognize Table 1 as y � x2,and they can use this table as a basis for othercomparisons.

• For Table 2, students may see that each number in they column is 100 more than the square of the numberin the x column. Therefore the equation is y � x2 �

100.

• For Table 7, students may see that each number in they column is a perfect square. For example, 36 is 62,so the question now becomes how to get 6 from �3.Multiplying �3 by �2 gives 6, so the equation mightbe y � (�2x)2, or y � 4x2.

Extra Help If students are having trouble findingequations, you may want to supply a list of possible correctand incorrect equations for them to choose from. Your listmay look something like the following:

y � 4x2 y � 4x2� 4 y � x2 � 100

y � 2x2 y � x2 y � x2 � 4

y � 4x2 � 1 y � 2x2 � 2 y � x2 � 20

y � 10x2 y � 4x2 � 1 y � x2 � 4

y � x2 � 10 y � 10x2 � 20

Students seem to enjoy trying to find equations for datagiven in tables. Encourage them to share their problem-solving strategies for finding these equations; some ofthem may surprise you!

Share & SummarizeThese questions are a good assessment of whetherstudents see the connections between graphs andequations and between tables and equations.

In Question 1, encourage students to compare thegraphs. For example, they might think about how they values in Graph B relate to the y values in Graph Afor a particular x value.

Problem-Solving Strategies


Develop

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&


Problem Set B

For the tables in Problems 1–7, find a quadratic equation. Hint: ForProblems 2–7, compare the values with those in the tables from previousproblems.

1. x y 2. x y 3. x y 4. x y�3 9 �3 109 �3 5 �3 13�2 4 �2 104 �2 0 �2 8�1 1 �1 101 �1 �3 �1 5

0 0 0 100 0 �4 0 4

1 1 1 101 1 �3 1 5

2 4 2 104 2 0 2 8

3 9 3 109 3 5 3 13

5. x y 6. x y 7. x y 8. x y�3 90 �3 70 �3 36 �3 37�2 40 �2 20 �2 16 �2 17�1 10 �1 �10 �1 4 �1 5

0 0 0 �20 0 0 0 1

1 10 1 �10 1 4 1 5

2 40 2 20 2 16 2 17

3 90 3 70 3 36 3 37

ShareSummarize

1. Graph A is the graph of y � x2. Write equations for the othergraphs, and explain how you know the equations fit the graphs.

4

6

�4

�4

y

x

Graph A

4

6

�4

�4

y

x

Graph B

4

6

�4

�4

y

x

Graph C

1. y � x2

2. y � x2 � 1003. y � x2 � 44. y � x2 � 4

5. y � 10x2

6. y � 10x2 � 207. y � 4x2

8. y � 4x2 � 1

1. Graph B:y � x2 � 2; it hasthe same shape asy � x2 but passesthrough the y-axis2 units higher.Graph C:y � x2 � 3; it looks the same asy � x2 but passesthrough the y-axis3 units lower.


➧1 You may have studentswork in pairs.

➧2 If students havetrouble, give them alist of equations tochoose from.

➧3 Have students sharetheir problem-solvingstrategies.

➧4 Encourage studentsto compare thegraphs.

082a-107a TE 02/2 MSMgr8 5/22/04 9:31 AM

In Question 2, encourage students to explain theirreasoning, for example:

In Table C, you get each y value by taking eachx value, subtracting 2, and squaring the result.

Troubleshooting Students will get much more practicewith these types of problems in the next investigation. Itis fine to go on from here even if they haven’t masteredthe connection between equations and graphs ofquadratic relationships.


Practice & Apply: 1–7, pp. 98–99Connect & Extend: 26, p. 103Mixed Review: 32–44, pp. 106–107

Investigation 2In this investigation, students use a graphing calculatorto explore the effects of a, b, and c on equations of theform y � ax2 � bx � c. Students should develop moregeneral graphing sense about constructing parabolas,including their distinctive points (high and low points,and intercepts).

So that graphs can be drawn quickly, students shouldwork with a partner and share a graphing calculator tocarry out this investigation. Working with a partner alsoallows students to discuss their responses before writingdown their observations and conclusions.


Develop

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Investigation

2. Table A represents the relationship y � x2. Write the quadraticequations represented by Tables B and C. Explain how you foundyour equations.

Table A Table B Table C

x y x y x y�4 16 �4 �48 �4 36�3 9 �3 �27 �3 25�2 4 �2 �12 �2 16�1 1 �1 �3 �1 9

0 0 0 0 0 4

1 1 1 �3 1 1

2 4 2 �12 2 0

3 9 3 �27 3 1

4 16 4 �48 4 4

2 Quadratic Equationsand Their Graphs

Quadratic equations can be written in the form y � ax2 � bx � c, wherea, b, and c are constants and a is not 0. Like the coefficient m and con-stant b in a linear equation y � mx � b, the coefficients and constant in aquadratic equation tell you something about the graph of the equation.

In Problem Set C, you will explore how the coefficient a and constant caffect the graphs of quadratic equations. In Problem Set D, you will lookat the effect of b, which isn’t as easy to see.

2. Table B representsy � �3x2 sinceeach y value is �3times the squareof the x value.Table C representsy � (x � 2)2 sinceeach y value is thesquare of 2 lessthan the x value.

The cables of a suspen-sion bridge form theshape of a parabola.

factsJust

t h e



➧1 Encourage studentsto explain theirreasoning.

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MathematicsAbout t h e


When graphing the groups of equations in Problem Sets Cand D, students can easily get the impression that there area number of shapes of quadratic graphs. For example, thegraphs of y � x2 � 1 and y � 2x2 � 1 may seem to bedifferent shapes when graphed in the same window. Butthere is only one parabolic shape; these partial parabolasonly look “wider” or “thinner” because less or more,respectively, of the graph is showing.

This property is not true of all graphs; in Investigation 4,students will see graphs of cubic functions, which can bemuch more pronounced.

Problem Set C Suggested Grouping: PairsIn this problem set, students graph groups of quadraticequations in the form y � ax2 � c. The equations ineach group vary a or c (but not both) so that studentscan easily see how each affects the graph.

Before students beginworking, you might handout Master 43, graphingcalculator grids, for thisproblem set as well asProblem Set D. You maywant to go over withstudents how to properlylabel a sketch made from agraphing calculator: drawin the tick marks, label thefirst and last tick marks oneach axis, and add labelsto the axes for the vari-ables. By labeling only the end points, students caneasily tell what the scale is,and the graphs won’t becluttered with tick-mark labels.

3

5

�3 �1

In Problem 1, students should work in pairs on onegroup of equations at time. Encourage partners to lookfor as many similarities as possible (high and lowpoints, width, direction the parabolas open, mirrorimages) and to share their observations in Problems 1b,2, and 3.

Groups such as those in Problem 1 are often referred toas “families” of functions. The word families here isappropriate since these equations have characteristics incommon.

Problems 4 and 5 assess whether students are cor-rectly identifying the effects of a and c. It is importantthat students understand that in all of the problems theyhave graphed until this point, b has been equal to 0. Itwill become apparent in Problem Set D, when studentsare given quadratic equations with b not equal to 0,that it is much easier to analyze the effects of a and cwhen b is 0.

Problem Set Wrap-Up Review Problems 4 and 5with the class. If some students are confused, you maywant to discuss what clues students should have seenin Problems 1–3 to help them answer these last twoproblems. Use this discussion to lead into the twoparagraphs following the problem set. ■

Discuss the information at the bottom of the page, makingsure students understand that if a quadratic equation isin the form y � ax2 � c, the highest or lowest point is atthe point (0, c). You can use this opportunity to introducethe term vertex for the highest or lowest point of aparabola. Explain that the vertex of a parabola is animportant point, since in many applications of quadraticequations, you want to know the maximum or minimumvalue. Students should become comfortable using thisterm as a way to describe the maximum or minimumpoint of a parabola.


Develop

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Teaching Resources

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Problem Set C

1. Complete Parts a–c for each group of quadratic equations.

Group I Group II Group III Group IV

y � x2 y � x2 y � x2 y � �x2

y � x2 � 1 y � x2 � 1 y � 2x2 y � �2x2

y � x2 � 3 y � x2 � 3 y � �12

�x2 y � ��12

�x2

a. Graph the three equations in the same window of your calculator.Choose a window that shows all three graphs clearly. Make asketch of the graphs. Label the minimum and maximum values oneach axis. Also label each graph with its equation.

b. For each group of equations, write a sentence or two about howthe graphs are similar and how they are different.

c. For each group of equations, give one more quadratic equationthat also belongs in that group.

2. Describe how the graphs in Group I are like the graphs in Group IIand how they are different.

3. Describe how the graphs in Group III are like the graphs inGroup IV and how they are different.

4. Use what you learned in Problems 1–3 to predict what the graphof each equation below will look like. Make a quick sketch of thegraphs on the same set of axes. Be sure to label the axes, and labeleach graph with its equation. Check your predictions with yourcalculator.

a. y � x2 � 2

b. y � 3x2 � 2

c. y � �3x2 � 2

d. y � �12

�x2 � 3

5. All the equations you looked at in this problem set are in the formy � ax2 � bx � c, but the coefficient b is equal to 0. Explain howthe values of a and c affect the graph of an equation.

In Problem Set C, you probably saw that equations of the form y � ax2 � chave their highest or lowest point at the point (0, c). The highest or lowestpoint of a parabola is called its vertex.

Not all parabolas have their vertices on the y-axis. In the next problemset, you will look at the properties of an equation that determine wherethe vertex of its graph will be.

See below.

M A T E R I A L Sgraphing calculator

1a, b. See AdditionalAnswers.

1c. Possible answer:Group I: y � x2 �4; Group II: y �x2 � 4; Group III:y � 7x2; GroupIV: y � �7x2

2. Graphs in bothgroups areparabolas that openupward. The graphsin Group I cross they-axis at or above(0, 0), while thegraphs in Group IIcross the y-axis ator below (0, 0).

3. Possible answer:The graphs inGroup III openupward while thegraphs in Group IVopen downward.The graph of eachequation in GroupIV is the mirrorimage of the graphin Group III with theopposite coefficient.

V O C A B U L A R Yvertex

5. If a is positive, the graph opens upward. If a is negative, the graph opens downward. Thegreater the absolute value of a, the narrower the graph. The value of c determines where thegraph crosses the y-axis.

y � x2 � 3

y � x2 � 2

y � �3x2 � 2

y � 3x2 � 2

4

7

�4

�5

12


Additional Answers1a. See page A364.

1b. Possible answer:Group I: The graphs are parabolas that open upward, but they cross the y-axis at different points: (0, 0), (0, 1), and (0, 3), respectively.Group II: The graphs are parabolas that open upward, but they cross the y-axis at different points: (0, 0), (0, �1), and (0, �3), respectively.Group III: The graphs are parabolas that open upward and have their lowest point at (0, 0), but they have different widths. The narrowest isy � 2x2 and the widest is y � �

12

�x2.Group IV: The graphs are parabolas that open downward and have their highest point at (0, 0), but they have different widths. The narrowestis y � �2x2 and the widest is y � ��

12

�x2.


➧2 Discuss how to makea sketch of a graphingcalculator graph.

➧3 Discuss Problems 4and 5.

➧4 Introduce the newvocabulary term.

082a-107a TE 02/2 MSMgr8 5/22/04 9:33 AM



Problem Set D Suggested Grouping: PairsIn this problem set, students continue to comparequadratic equations, but within each group only thecoefficient b varies.

In Problem 1, students may hypothesize that the vertex(rather than the y-intercept) will be at the point (0, c),because for each parabola in Problem Set C, the vertexwas the y-intercept. When they check their predictions inProblem 2b, you may be able to take the opportunityto point out that while the vertex moved from the point(0, c), the y-intercept did not—so they have still noticedsomething important: that the point (0, c) is on the graph.

In Problem 2, students may begin to notice that thex term in the quadratic equation seems to have the effectof moving the parabola up or down and right or left.Students will make use of this observation in Problem 3.

Problem 3 should be answered in very general terms.While it is possible to locate the vertex of parabolas iny � ax2 � c form, when the equation has an x term(b is not 0), the vertex is more difficult to identify fromthe equation. Later in the course, students will seeparabolas in another general form, y � a(x � h)2 � k,a form that allows students to identify the vertex readily:the point (h, k). At this stage, students are expected onlyto see that b will move the parabola, not exactly how itwill move it. They should also understand that if b is not0, the parabola will not have its vertex on the y-axis.

Problem Set Wrap-Up Discuss Problem 1 as a class.In each group of equations, the values for a and c arethe same, so the graphs should have the same widthand y-intercept. You can then ask students:

If the width of a particular parabola has to stay thesame, and the graphs have to cross the y-axis atthe same place, how can the parabola change?

You can use a piece of wire shaped like aparabola, or a trans-parency with a paraboladrawn on it, to demon-strate just how it mightchange: slide the parabolay � ax2 � c through they-intercept, retaining theorientation.

10

8

6

4

2

�2�6 �4 �2 642

y

x

The parabolas move up or down in the opposite directionfrom how they open (when a is positive they move down,when a is negative they move up) as b increases ordecreases from 0. They also move left or right dependingon the sign of a and whether b is increasing or decreasing:when a is positive, they move left for b increasing from 0and right for b decreasing from 0; when a is negative, thepattern is right for b increasing and left for b decreasing.

Extra Challenge Give students the following equations,and ask them to find the line of symmetry (or the vertex) ineach case. (All occur at x � 1.) Some students may beable to find a connection between the equations and theline of symmetry, or between the equations and the x-coordinate of the vertex.

y � x2 � 2x y � 2x2 � 4x

y � �3x2 � 6x y � x2 � 2x � 1

y � 2x2 � 4x � 1 y � �3x2 � 6x � 1

Share & SummarizeAfter students have written answers to these questions, inpairs or individually, discuss them as a class. You maywant to ask how students found their answers.

Question 1 encourages students to think about howthe value of c affects the graph of the parabola. Thephrase “without changing its shape” is key: there wouldbe an infinite number of possible equations if the shapecould vary, since a could be any number except 0.

In Question 2, you may want to discuss with studentswhat making a “rough sketch” means. Clearly, the ver-tex must be included and an indication of whether theparabola opens up or down, and some reasonable esti-mate of width (perhaps by plotting one or two points oneither side of the line of symmetry).


Develop

• Teaching notes continued on page A364

➧1

➧2

➧3 ➧4

082a-107a TE 02/2 MSMgr8 5/22/04 9:33 AM

&


Problem Set D

Consider these four groups of quadratic equations.

Group I Group II Group III Group IV

y � x2 � 2 y � x2 � 2 y � �2x2 � 2 y � �2x2 � 2

y � x2 � 3x � 2 y � x2 � 3x � 2 y � �2x2 � 3x � 2 y � �2x2 � 3x � 2

y � x2 � 6x � 2 y � x2 � 6x � 2 y � �2x2 � 6x � 2 y � �2x2 � 6x � 2

1. In each group, all the equations have the same values of a and c.Use what you learned about the effects of a and c to make predic-tions about how the graphs in each group will be alike.

2. For each group of equations, complete Parts a–c.

a. Graph the three equations in the same window of your calculator.Choose a window that shows all three graphs clearly. Make asketch of the graphs. Remember to label the axes, and label thegraphs with their equations.

b. Were your predictions in Problem 1 correct?

c. For each group of equations, write a sentence or two about howthe graphs are similar and how they are different.

3. What patterns can you see in how the locations of the parabolaschange as b increases or decreases from 0?

ShareSummarize

1. Imagine moving the graph of y � �2x2 � 3 up 2 units withoutchanging its shape. What would the equation of the newparabola be?

2. Briefly describe or make a rough sketch of the graph ofy � ��

12

�x2 � 2.

3. For each quadratic equation, tell whether the vertex is on they-axis.

a. y � �12

�x2 � 3 b. y � x2 � 3x � 1

c. y � �x2 d. y � �3x2 � x � 13

e. y � �x2 � 4 f. y � 7x2 � 3x noyes

noyes

noyes

y � �2x2 � 5

Answers will vary.


1. The graphs in eachgroup will openin the same directionand have the samewidth; they willalso have the samey-intercept.

2a, c. See AdditionalAnswers.

3. Possible answer:The parabolas movefarther from theorigin. In Group I,the graphs havedifferent vertices,moving left anddown as bincreases. In GroupII, the graphs havedifferent vertices,moving rightand down as bdecreases. In GroupIII, the graphs havedifferent vertices,moving rightand up as bincreases. In GroupIV, the graphs havedifferent vertices,moving left and upas b decreases.

Share & Summarize Answer

2. It is a parabola opening downward with its vertex at (0, �2); it willbe wider than the graph of y � x2.


Additional AnswersProblem Set D2a. See page A365.

2c. Group I: The graphs are parabolas that open upward, seem to be about the same width, and have the same y-intercept.Group II: The graphs are parabolas that open upward, seem to be about the same width, and have the same y-intercept.Group III: The graphs are parabolas that open downward, seem to be about the same width, and have the same y-intercept.Group IV: The graphs are parabolas that open downward, seem to be about the same width, and have the same y-intercept.For each group, the vertex of each graph is different.


➧2 Discuss the locationof the vertex.

➧3 Discuss Problem 1as a class, perhapsdemonstrating witha piece of wire.

➧4 Have students writetheir answers, andthen discuss themand students’strategies.

➧5 Offer additionalactivities for studentshaving trouble seeingthe effects of a and c.

082a-107a TE 02/2 MSMgr8 5/22/04 9:34 AM

Investigation 3In this investigation, students explore how quadraticequations and graphs relate to the motion of objects thatare thrown or shot into the air. Describing the path, ortrajectory, of such objects is the most common applicationof parabolic motion. A quadratic equation can be usedto approximate the relationship between a ball’s heightand the horizontal distance it travels.

One important point to note here is that students are firstintroduced to trajectories through equations that relatethe height of an object to the time it has traveled; thegraph is of height versus time. In contrast, the graph ofthe actual trajectory describes the relationship betweenthe height of an object and its horizontal distance alongthe ground. Students often mistakenly read height-versus-time graphs as though they were height-versus-horizontal-distance graphs, which are actually pictures of anobject’s motion.

Introduce the investigation by asking students to watchcarefully and then tossing a ball or wad of paper fromone end of the room to the other. Toss it so that thehighest point it reaches is just below the ceiling. (Tossingunderhand is best, and a bit of practice is helpful.)Students will see that if the ball is launched at an anglebetween 0 and 90 degrees, its path, or trajectory,approximates the shape of a parabola.

Problem Set E Suggested Grouping: PairsIn this problem set, students examine a graph that showsthe height over time of an arrow shot into the air. Besure students understand that this is a graph of heightversus time, not the graph of the actual path the arrowtook. This problem set should take no more than 5 to10 minutes.


Develop

➧1

➧2➧3

082a-107a TE 02/2 MSMgr8 5/22/04 9:34 AM

Investigation


3 Using Quadratic Relationships

Quadratic equations and graphs can help you understand the motion ofobjects that are thrown or shot into the air. If an object, like a ball, islaunched at an angle between 0° and 90° (not straight up or down), itstrajectory—the path it follows—approximates the shape of a parabola.

This means that to approximate the relationship between the object’sheight and the horizontal distance it travels, you can use a quadratic equa-tion. As you will see in Problem Set A, there is also a quadratic relation-ship between the length of time the object is in the air and the object’sheight.

Problem Set E

A photographer set up a camera to take pictures of an arrow that had beenshot into the air. The camera took a picture every half second. The pointson the graph show the height of the arrow in meters each half secondafter the camera started taking photos. A parabola has been drawn throughthe points.

�1 0 1 2 3 4 5 6

Hei

ght

(m)

Time after Picture-taking Began (s)

50

40

30

20

10

Arrow Flight

Distance

Height


➧1 Introduce the investi-gation by tossing anobject into the airand having studentswatch its trajectory.


➧3 Make sure studentsunderstand that thisgraph shows heightversus time.

082a-107a TE 02/2 MSMgr8 5/22/04 9:35 AM

AssessmentOn t h e Spot

Problem 1 requires students to use the graph to interpretquestions about starting time and total time elapsed.Watch for students who, in Part b, think they should belooking at the entire span of time, rather than the timevalue corresponding to the rightmost point on the graphwith a height value of 24 meters.

In Problem 2, it is important that students be able tointerpret what it means about the height of the arrow if ithits the ground. (If it hits the ground, the height at thatpoint is 0.) This is a concept they will use often in thefuture.

Problem 4 asks students to estimate the total flight time.

Students may havevarious ways of approaching Problem 4.

• Some students may extend the graph to the left,noting that the intercepts are �1 and 5, which meansthe arrow was in the air for a total of 6 seconds.

• Other students may add their answers from Problems 2and 3 (5.1 and 0.9) to find a total time of 6 seconds.

Problem Set F Suggested Grouping: PairsIn this problem set, students work with a quadraticequation relating the height of a football to the horizontaldistance it travels. The height-versus-distance graphstudents make in Problem 2 reflects the actualtrajectory of the football.

Students may find it puzzling that the equation is notwritten in “standard” form with the x2 term first. Therefore,they may answer Problem 1 incorrectly if they think thesign of “2” will determine whether the parabola opens upor down. You may want to ask them to identify a, b, and cusing language such as “coefficient of x2” for a. It mayalso help to ask students to first rewrite the equation in theform y � ax2 � bx � c.



Develop

➧1

➧2

➧3

082a-107a TE 02/2 MSMgr8 5/22/04 9:36 AM

4. approximately 6 s;Possible explana-tion: Add theanswers toProblems 2 and 3.

Problem Set EAnswers

5a. 46 m; Find theheight value atthe highest pointon the graph.

5b. about 2.9 s; Addthe time it was inthe air before thecamera startedshooting (0.9 s) tothe time it took toreach its maximumheight (2.0 s).


1. Consider the arrow’s height when the camera took the first picture.

a. How high was the arrow at that time? Explain how you know.

b. How much time elapsed from that point until the arrow returned to that height? Explain how you know.

2. How long after the first photo was taken did the arrow hitthe ground? How did you determine this?

3. The arrow was shot from a height of 1.5 m (just above shoulder height). About how long before the time of the first photo was the arrow shot? How did you determine this?

4. Approximately how long was the arrow in flight? Explain how you found your answer.

5. Now think about how high the arrow rose.

a. What was the arrow’s greatest height? Explain how you foundyour answer.

b. How many seconds after the arrow was shot did it reach its maximum height? Explain how you found your answer.

Problem Set F

A quarterback threw a pass in such a way that the relationship between itsheight y and the horizontal distance it traveled x could be described bythis equation:

y � 2 � 0.8x � 0.02x2

Both y and x are measured in yards.

1. Will the graph of this quadratic equation open upward or down-ward? Explain.

2. Use your calculator to graph the equation. Use a window that givesa view of the graph for the entire time the ball was in the air. Youcan change the window settings by adjusting the minimum and max-imum x and y values for the axes and the x and y scales. For thisproblem, try an x scale of 10 and a y scale of 2. Sketch the graph,being sure to label the axes. See Additional Answers.

Downward; the coefficient of x2 is negative.

1a. 24 m; This is theheight on the graphcorresponding to atime of 0 s.

1b. 4.2 s; This is thetime value corre-sponding to theother 24-m heightvalue on the graph.

2. 5.1 s; Find thetime at the pointwhere the height is0 (the ground).

3. about 0.9 s; Findthe first point of thegraph where theheight is 1.5 m.This correspondsto a time value of�0.9 s, which is0.9 s beforefilming began.



Additional AnswersProblem Set F2.

50

12

0

➧1 Make sure studentsare considering theentire elapsed time.


➧3 Watch for studentswho answer thisquestion incorrectly.

082a-107a TE 02/2 MSMgr8 5/22/04 9:36 AM

In Problem 3, students need to figure out that the heightfrom which the football was thrown is at the point wherex � 0.

In Problem 5, it might be useful to point out to studentsthat the ball reaches a height of 2 yards (the height fromwhich it was first thrown) twice. This is an importantfeature of trajectories, in that projectiles can reachspecific heights at two times, one on the way up andone on the way down.

Share & SummarizeYou may want to talk about the symmetry of the parabolawhen discussing Question 2. Ask students where theline of symmetry is for this parabola, and encouragethem to think about mirror-image points.

Troubleshooting If students are having difficultywith these problems, you can still continue to the nextinvestigation. Students will get more practice applyingquadratics (as well as other types of equations) in laterchapters.


Practice & Apply: 13, p. 101Connect & Extend: 29, pp. 104–105Mixed Review: 32–44, pp. 106–107


Develop

➧1

➧2

082a-107a TE 02/2 MSMgr8 5/22/04 9:37 AM

&


4. 10 yd; this is thegreatest y valueon the graph, whenx � 20.

5. 40 yd; At x � 0,the height was 2 yd.The height is 2 ydagain at x � 40.

1. Find the highestpoint on the graph,and estimate thefirst coordinate,which gives the time.

For Problems 3–5, use your calculator graph or the equation to answerthe question.

3. From what height was the football thrown? Explain how you foundyour answer.

4. What was the greatest height reached by the ball? Explain how youfound your answer.

5. A receiver caught the ball in the end zone at the same height it wasthrown. What distance did the pass cover before it was caught?Explain how you found your answer.

ShareSummarize

Marcus hit a tennis ball into the air. The graph shows the ball’s height h infeet over time t in seconds. The point P shows the ball’s height at t � 1.

1. Explain how to use the graph to estimate when the ball reached itshighest point.

2. Explain how to use the graph to find another time when the ballhad the same height it had at t � 1.Possible explanation: Draw or imagine a horizontal linethrough Point P. Find the other intersection of this line and theparabola; estimate its first coordinate, which is the time whenthe ball had the same height it had at t � 1.

80

70

60

50

40

30

20

10

h

t

0 1 2 3 4 5

P

Time(s)

Hei

ght

(ft)

Tennis Ball Height

2 yd; this is the y value when x � 0.


➧1 Point out that theball is at a height of2 yards twice.

➧2 Talk about thesymmetry of theparabola.

082a-107a TE 02/2 MSMgr8 5/22/04 9:37 AM

Investigation 4In this investigation, students extend their knowledge offamilies of functions to cubics and compare the charac-teristics of linear, quadratic, and cubic functions.

Problem Set G Suggested Grouping: Pairs or Individuals

In this problem set, students are asked to identifyquadratic relationships from their symbolic forms. This isan important problem set, because it shows students avariety of ways to express quadratic relationships andprovides a good review of algebraic skills in manipu-lating expressions.

Students should recognize quadratic equations in“standard” form, as in Problems 1, 9, and 10.Although they haven’t concentrated on quadratics in thesame form as Problems 2 and 3, they did see thisform in the first tables of Investigation 1. You may wantto remind them of those tables. In a later chapter, theywill see more quadratics in this form.

Problem 5 may cause some confusion; students maythink that since the equation has an x2 term, it isquadratic. Point out that the definition of a quadraticrelationship does not include terms with x2 in thedenominator. In Chapter 3, they will learn that �

x12� is

equivalent to x�2 and that dividing by x2 is equivalentto multiplying by x�2.

Students may think the equation in Problem 8 isquadratic because it has a squared term, 2x2. Point outthat for an equation to be quadratic, the highest powermust be 2.

After students complete these problems, define acubic equation as one that can be written in the formy � ax3 � bx2 � cx � d. Ask students to find the cubicequations in the text and identify a, b, c, and d foreach. This is good practice for when they encounter thequadratic formula in Chapter 7. Explain to students thatin the next few problem sets, they will be looking at thecharacteristics of graphs of cubics.


Develop

➧1

➧2

➧3

082a-107a TE 02/2 MSMgr8 5/22/04 9:38 AM

Investigation


4 Comparing Quadraticsand Other Relationships

These problems will help you distinguish between quadratic relationshipsand other types of relationships.

Problem Set G

One way to distinguish different types of relationships is to examinetheir equations.

Tell whether each equation is quadratic. If an equation is not quadratic,explain how you know.

1. y � 3m2 � 2m � 7 2. y � (x � 3)2 � 7

3. y � (x � 2)2 4. y � 10

5. y � �x22� 6. y � b(b � 1)

7. y � 2x 8. y � 3x3 � 2x2 � 3

9. y � �2.5x2 10. y � 4n2 � 7n

11. y � 7p � 3 12. y � n(n2 � 3)

Some of the equations in Problem Set G are cubic equations. A cubicequation can be written in the form y � ax3 � bx2 � cx � d, where a ≠ 0.These are all examples of cubic equations:

y � x3 y � 2x3 y � 0.5x3 � x � 3 y � x3 � 2x2

The graphs of cubic equations have their own characteristics, differentfrom those of linear and quadratic equations.

V O C A B U L A R Ycubic equation

1. quadratic2. quadratic3. quadratic4. not quadratic; no

squared variable5. not quadratic; the

constant is divid-ed, rather thanmultiplied, by x2

6. quadratic7. not quadratic; no

squared variable8. not quadratic;

contains a cubedvariable

9. quadratic10. quadratic

12. not quadratic; ifexpanded, wouldinclude a cubedvariable

11. not quadratic; nosquared variable

Cubic equations areused by computerdesign programs tohelp draw curved linesand surfaces.

factsJust

t h e


➧1 You may have studentswork in pairs or ontheir own.

➧2 • For Problems 2 and3, remind studentsof the tables inInvestigation 1.

• For Problem 5,mention that thedefinition of aquadratic does notinclude x2 terms inthe denominator.

• For Problem 8,remind studentsthat the highestexponent must be 2.

➧3 Define and reviewexamples of cubicequations.

082a-107a TE 02/2 MSMgr8 5/22/04 9:38 AM

Problem Set H Suggested Grouping: Pairs or Individuals

Problem 1 forms the basis for much of the comparisonwork students will do in graphing. They should becomevery familiar with these basic curves and their charac-teristics. Encourage them to talk about symmetry,vertices, intercepts, and how y behaves as x increasesor decreases.

In Investigation 1, students considered the effects of aand c on graphs of quadratic equations in the formy � ax2 � c. In Problem 2, they will look at simplecubic equations in the form y � ax3 � d to consider theeffects of a and d. Encourage students to refer to linesand parabolas when talking about the effects of theseconstants. In lines, as the coefficient of x increases, theline becomes steeper; in parabolas, as the coefficient ofx2 increases, the parabola becomes thinner. You maywant to have students predict what will happen withgraphs of cubics as a increases.

Likewise, students should note the similarities to whathappens with parabolas when c is changed.


Develop

➧1

➧2

➧3

082a-107a TE 02/2 MSMgr8 5/22/04 9:39 AM


Problem Set H

1. Consider these three equations.

y � x y � x2 y � x3

a. Graph the three equations in the same window of your calculator.Choose a window that shows all three graphs clearly. Sketch thegraphs, and remember to label the axes and the graphs.

b. Write a sentence or two about how the graphs are the same andhow they are different.

c. Give the coordinates of the two points where all three graphsintersect.

You have seen how the values of m and b affect the graph of y � mx � band how the values of a, b, and c affect the graph of y � ax2 � bx � c.

You will now consider simple cubic equations of the form y � ax3 � d tosee how changing the coefficient a and the constant d affect the graphs.

2. Complete Parts a–c for each of these three groups of equations.

Group I Group II Group III

y � x3 y � 2x3 y � 3x3 � 1

y � x3 � 3 y � �12

�x3 y � 3x3 � 1

y � x3 � 3 y � �2x3 y � �3x3 � 1

a. Graph the three equations in the same window of your calculator.Choose a window that shows all three graphs clearly. Sketch andlabel the graphs.


c. What does the coefficient of the x3 term seem to tell you aboutthe graph?

d. What does the constant tell you about the graph?

See Additional Answers.

See below.

(0, 0) and (1, 1)


1a. See AdditionalAnswers.

1b. All three passthrough (0, 0).The first graph is aline; the others arecurves. The secondgraph has a lowpoint at (0, 0).The third graphchanges thedirection it curvesat (0, 0). Thesecond graph issymmetric aboutthe y-axis; theothers aren’t. Inthe first and thirdgraphs, the yvalues increasefrom left to right. Inthe second graph,the y valuesdecrease until(0, 0), and thenincrease.

The words linear, qua-dratic, and cubic comefrom the idea thatlines, squares (alsocalled quadrangles),and cubes have 1, 2,and 3 dimensions,respectively.

factsJust

t h e

2c. how wide the graph is, and whether it goes up or down from leftto right

2d. the y-intercept, or whether the graph is moved up or down fromthe origin

2a. y � x3

y � x3 � 3

y � x3 � 3

3

10

�3

�10

Group Iy � 2x3

y � �2x3

3

10

�3

�10

Group II y � x312

y � 3x3 � 1

y � �3x3 � 1

10

�10

Group III y � 3x3 � 1

3�3


Additional Answers1a.

y � x

y � x3y � x2

4

4

�4

�4

2b. Group I: They have the same shape but different y-intercepts.Group II: All three go through the origin, but the one with the �

12

� coefficient is wider. The onewith the �2 coefficient goes down from left to right; the other two go up from left to right.Group III: They’re all the same width and shape. The third equation has a graph that goesdown from left to right. The second and third go through (0, �1); the first goes through (0, 1).


➧2 Encourage studentsto discuss symmetry,vertices, intercepts,and how y changes asx changes.

➧3 Encourage studentsto discuss lines andparabolas whenexamining the effectsof the constants.

082a-107a TE 02/2 MSMgr8 5/22/04 9:39 AM


Problem Set I Suggested Grouping: Pairs or Individuals

In this problem set, students look at the effects of thex2 and x terms on the shape of a cubic. They may besurprised that the graphs of the cubics in this problemset have different shapes from those in the general formy � ax3 � d, which they saw in Problem Set H.

In Problem 2, you may want to ask students why theythink the first and third graphs go up, then down, thenup, while the middle graph goes down, then up, thendown. They might correctly conjecture that the sign of adetermines the order in which they go up and down.

Extra Challenge This is a good opportunity for studentsto do some analysis of polynomial behavior for values ofx far from 0. Ask students to write a paragraph or two toexplain what happens to y � x3 � x as x gets far from 0,either positive or negative. They may see that as positive xincreases, x3 increases very quickly and �x becomes lesssignificant. Therefore, when x gets very far from 0, x3 � xwill also get very far from 0. Likewise, for negative anddecreasing x, x3 will decrease more quickly than �xincreases.

This kind of analysis will help students predict the shape acubic will assume just by looking at the equation.

In Problem 3, most students will be able to see wherethe x-intercepts are, but those who are familiar with theircalculators may want to use the Trace feature to findthem. Either approach is fine at this point. (See pageT242 of Chapter 4 or Master 14 for instructions onhow to use the Trace feature.) To find the y-intercepts,however, encourage students to look at the equationsand mentally determine the value of y when x is 0.

In Problem 4, students should begin to generalize thatadding a constant onto the equation has the effect ofmoving the graph vertically, up if it is positive and downif it is negative.

Some students may usetheir graphing calculators to experiment with equations.Encourage this kind of experimentation wheneverpossible so students can test their conjectures. Otherstudents may use their previous knowledge of parabolasto answer this question. Both approaches are fine.


Share & SummarizeThis question allows students to begin to make general-izations about the graphs of linear, quadratic, and cubicequations. Students might confer in small groups or pairsabout their responses and then share them with theclass. It is also helpful to make a large chart on abulletin board or other relatively permanent spot thatoutlines the characteristics students identify.

Troubleshooting If students are having difficulty inanalyzing cubic equations and their graphs, go back tothe simplest forms (y � ax3 and y � ax3 � d ) and havestudents invent and graph their own equations. Thenhave them take each of their equations and add an x orx2 term to see the effects on the shape of the graph.


Practice & Apply: 14–25, p. 102Connect & Extend: 30, 31, pp. 105–106Mixed Review: 32–44, pp. 106–107


Develop

➧1

➧2

➧3

➧4

➧5

082a-107a TE 02/2 MSMgr8 5/22/04 9:40 AM

&


Problem Set I

Now consider what happens when a cubic equation has x2 and x terms.Consider these three equations.

y � x3 � x y � �x3 � 2x2 � 5x � 6 y � 2x3 � x2 � 5x � 2

1. Graph the equations in the same window of your calculator, andsketch the graphs. Be sure to label the graphs and the axes.

2. Describe in words the general shapes of the graphs.

3. For each graph, find the points where the graph crosses the x-axisand y-axis.

4. Suppose you moved the graph of y � x3 � x up 1 unit.

a. What is the equation of the new graph?

b. Graph your new equation. How many x-intercepts does the graphhave?

ShareSummarize

Look back at the graphs you made for this investigation. Describe whatyou observe about how graphs of linear, quadratic, and cubic relation-ships differ.

See Additional Answers; 1

y � x3 � x � 1


1, 3. See AdditionalAnswers.

2. The graphs areall curves thatchange directiontwice. Looking fromleft to right, thegraphs with positivex3 terms go up, thendown, then upagain. The graphwith a negative x3

term goes down,then up, then downagain.

Cubic equations areoften involved in thedesign of the curvedsections of sailboats.

factsJust

t h e

Possible answer: Graphs of linear relationships are lines.Graphs of quadratic relationships are symmetrical curves that look like bowls opening upward ordownward. Graphs of cubic relationships have one half that opens upward and one that opens down-ward. Graphs of linearequations don’t turn; thoseof quadratic equations turnonce; those of cubic equa-tions curve but don’t turn ifthere is no x2 or x term,and turn twice if there is anx2 or x term.


Additional AnswersProblem Set I1. See page A365.

3. y � x3 � x: x-axis (�1, 0) (0, 0) (1, 0), y-axis (0, 0)y � �x3 � 2x2 � 5x � 6: x-axis (�2, 0) (1, 0) (3, 0), y-axis (0, �6)y � 2x3 � x2 � 5x � 2: x-axis (�1, 0) (��

12

�, 0) (2, 0) y-axis (0, �2)


➧2 Ask students whythe graphs have theshapes they do.

➧3 Encourage studentsto calculate they-intercepts mentally.

➧4 Have students conferin pairs or smallgroups before sharinganswers as a class.

➧5 Give students who arehaving trouble withcubics extra practice.

082a-107a TE 02/2 MSMgr8 5/22/04 9:40 AM

Grouping: Small GroupsThis lab gives students an opportunity to use theirknowledge of quadratic equations as they duplicatedesigns made of parabolas. Students should work insmall groups so that there can be some discussion andsharing of ideas about how to re-create these graphs.

Materials and PreparationThis lab investigation works best if each student has agraphing calculator. If you do not have enough calcu-lators for each student, each pair of students can shareone; use groups of four or five. Make sure, however,that students switch roles at some point so that everyonehas the experience of using the graphing calculator. Youmay want to have one student in each pair do the firstgraphing exercise (Try It Out) and the other student dothe second (Try It Again).

You may want to give copies of Master 43, graphingcalculator grids, to students to make sketching easier.

IntroduceIf students completed the Chapter 1 lab investigation,remind them of the linear designs they created in thatlab. Explain that, in this lab, they will use their knowl-edge of parabolas and their associated equations toduplicate some designs made from parabolas.

It is important to note that students are not being askedto look at specific points (other than vertices) to re-createthese graphs, nor do they need to worry that theirgraphs look exactly like those shown. The only criteriastudents need to consider are stated under each design.


Develop

LabInvestigation

➧1

➧2

▲ Name Date

Master43

© 2

001

Ever

yday

Lea

rnin

g C

orpo

ratio

n

T E A C H I N G R E S O U R C E S Additional Masters 59

Blank Calculator Graphs

Teaching Resources

082a-107a TE 02/2 MSMgr8 5/22/04 9:41 AM

LabInvestigation


Solving GraphDesign Puzzles

These simple designs are made from the graphs of quadratic equations.M A T E R I A L Sgraphing calculator Design A

In this design, the parabolas areequally spaced and have their ver-tices on the y-axis.

Design C

In this design, Parabola W has thesame width as Parabola Z, and theirvertices are the same distance fromthe origin. Parabola X has the samewidth as Parabola Y, and their ver-tices are the same distance fromthe origin.

Design B

In this design, each parabola has avertex at the origin. Parabola J hasthe same width as Parabola N.Parabola K has the same width asParabola M.

Design D

In this design, the vertices of thefour parabolas are equally spacedalong the x-axis. Two of thosepoints are to the left of the origin,and two are to the right.

J K

MN

W

Z

X

Y

RememberThe point where thex-axis and y-axis inter-sect, (0, 0), is calledthe origin.

You will use what you have learned about quadratic relationships to tryto re-create these designs on your calculator. Then you will create yourown design.


➧1 Have students work insmall groups.

➧2 Remind studentsof their work withlinear designs inthe Chapter 1 labinvestigation.

082a-107a TE 02/2 MSMgr8 5/22/04 9:41 AM

Try It OutMake sure students understand that they are to chooseone of Designs A, B, or C. (Design D is consideredlater.)

Each student should create his or her own design, sincefor Question 2, they will be comparing results withother members of the group. They are to enter equationsin the form y � ax2 � bx � c as they experiment withvalues of a, b, and c.

Some students may notice immediately that in all of thegraphs in Designs A, B, and C, the value of b must be 0since the graphs all have vertices on the y-axis. If theydo not remember this, do not tell them. The purpose ofthis lab is for them to experiment with different equations,trying to match the graphs. The beauty of graphingcalculators is that they allow students to perform exper-iments quickly and refine their conjectures as they gaininsight into the problems. If students continue to assigna value to b other than 0, you may want to give thema hint.

When a group thinks it has re-created one of thedesigns, check their answers and let them continue tothe next section if they are correct. Make sure they savetheir equations and sketches so that they compare themwith other groups’ equations at the end, if time permits.It is important that students notice that different equationsand windows can create the same designs if oneconsiders only the essential elements stated in thedescriptions.

Try It AgainStudents use the same process to create all of the otherdesigns, including Design D. Design D is the most diffi-cult because it requires students to think about assigninga nonzero value for b. However, the equations forDesign D can also be thought of in the form y � (x � c)2,for example, y � (x � 3)2, y � (x � 1)2, y � (x � 1)2,and y � (x � 3)2. For equations of this form, the lowestpoint on the graph touches the x-axis at (�c, 0). Studentswill consider this form more closely in Chapter 8.

If students are having a great deal of difficulty findingequations in the form y � ax2 � bx � c, you may wantto give them a hint about this alternative form, whichthey looked at briefly in the beginning of the lesson.Circulate around the room to check students’ finalgraphs and equations.

Take It FurtherStudents usually enjoy creating their own designs andchallenging other groups to re-create them. One obsta-cle may be that students create graphs that are too diffi-cult for others to duplicate. You may want to suggest thatthey think of an alternative, but in most cases, it wouldbe wise to give students an opportunity to take on thechallenge!

What Did You Learn?If time permits, have a whole-class discussion aboutstudents’ answers to Question 5 so students can get asense of the strategies other groups used.


Develop

➧2

➧1

➧3

➧4

➧5

082a-107a TE 02/2 MSMgr8 5/22/04 9:42 AM


Try It OutWith your group, choose one of Designs A, B, or C. Try to create thedesign on your calculator. Use equations in the form y � ax2 � bx � cand experiment with different values of a, b, and c. You may need toadjust the viewing window to make the design look the way you want.(Design D is the most difficult. You will have a chance to try it later.)

1. When you have created the design, make a sketch of the graph.Label each curve with its equation. Also label the axes, including themaximum and minimum values on each axis.

2. Different sets of equations and window settings can give the samedesign. Compare your results for Question 1 with the other membersof your group or with other groups who chose your design. Did yourecord the same equations and window settings?

Try It Again3. Now create each of the other three designs. For each design, make a

sketch and record the equations and window settings you used.

Take It Further4. Work with your group to create a new design from the graphs of

four quadratic equations.

a. Make a sketch of your design and—on a separate sheet ofpaper—record the equations and window settings you used.

b. Exchange designs with another group and try to re-create theirdesign.

What Did You Learn?5. Write a report about the strategies you used

to re-create the designs. For each design,discuss these points:

a. How did you change the coeffi-cients, a or b, or the constant, c, to create each design?

b. Did any of the coefficients orconstants have a value of 0? If so,

why was having that value equal to 0necessary to create the design?

c. Did you change the window settings to make any of the designs?If so, explain how changing either the range of x values or therange of y values affects the design.

Designs will vary.

3. See the answers forProblem 1.

1. See AdditionalAnswers.

2. Answers will vary.

The design of satellitedishes is based on theparabola.

factsJust

t h e

Reports will vary.


• Additional Answers on page A365

➧2 Allow students timeto realize that b = 0for Designs A, B, andC before giving thema hint.

➧1 Make sure studentschoose one ofDesigns A, B, or C.

➧3 Check students’designs.

➧4 Offer groups a hintabout Design D, ifnecessary.

➧5 Have groups sharetheir strategies.

082a-107a TE 02/2 MSMgr8 5/22/04 9:42 AM



1. For each table of values, find an equation it may represent. Look forconnections between the tables that may help you determine theequations. Explain how your found your solutions.

a. x y b. x y c. x y�3 9 �3 25 �3 0�2 4 �2 16 �2 1�1 1 �1 9 �1 4

0 0 0 4 0 9

1 1 1 1 1 16

2 4 2 0 2 25

3 9 3 1 3 36

d. x y e. x y f. x y�3 �18 �3 �6 �3 �8�2 �8 �2 �4 �2 2�1 �2 �1 �2 �1 8

0 0 0 0 0 10

1 �2 1 2 1 8

2 �8 2 4 2 2

3 �18 3 6 3 �8

In Exercises 2–5, match the equation with one of the graphs below.Explain your reasoning.

2. y � x2 3. y � (x � 2)2

4. y � (x � 3)2 5. y � �2x2

Graph A Graph D

Graph B

Graph C

x

y

&PracticeApply

1a. y � x2; The y values arethe squares of the xvalues.

1b. y � (x � 2)2; The yvalues are the squaresof the x values two rowsup, so they are found bysubtracting 2 from x andthen squaring.

1c. y � (x � 3)2; They values are the squaresof the x values threerows down, so they arefound by adding 3 to xand then squaring.

1d. y � �2x2; The y valuesare �2 times the y valuesfrom Table a.

1e. y � 2x; The y values are 2 times the corre-sponding x values.

1f. y � �2x2 � 10;The y values are10 more than the yvalues from Table d.

2. Graph D; its lineof symmetry is they-axis, and it opensupward.

3. Graph C; its line of symmetryis to the right of the origin, andits lowest point is to the right ofthe origin.

4. Graph A; its line of symmetryis to the left of the origin, andits lowest point is to the left ofthe origin.

5. Graph B; its line of symmetryis the y-axis, and it opensdownward.

98 C H A P T E R 2 Quadratic and Inverse Relationships impactmath.com/self_check_quiz

On YourOwnExercises

Investigation 1, pp. 84–87Practice & Apply: 1–7Connect & Extend: 26

Investigation 2, pp. 87–89Practice & Apply: 8–12Connect & Extend: 27, 28

Investigation 3, pp. 90–92Practice & Apply: 13Connect & Extend: 29

Investigation 4, pp. 93–95Practice & Apply: 14–25Connect & Extend: 30, 31

Assign AnytimeMixed Review: 32–44

Exercises 3 and 4:Watch for students whoreverse the direction of theparabolas in these exercis-es. Students may incorrect-ly think of y � (x � 2)2 asa shift 2 units to the leftrather than to the right,and of y � (x � 3)2 as ashift right rather than left.If so, ask them to thinkabout what value of x willmake y equal to 0.

AssessAssign and

082a-107a TE 02/2 MSMgr8 5/24/04 11:25 AM

http://impactmath.com/self_check_quiz

http://impactmath.com/self_check_quiz



In Exercises 6 and 7, match the equation with one of the graphs below.Explain your reasoning.

6. y � �2x2 � 2x � 3

7. y � 2x2 � 2x � 3

8. Could this be the graph of the equation y � �x2 � 1? Explain.

9. Could this be the graph of the equation y � �x2 � 1? Explain.

x

y

x

y

Graph A

Graph Bx

y

6. Graph B; the numbermultiplied by x2 isnegative, so the graphmust open downward.

7. Graph A; the numbermultiplied by x2 ispositive, so the graphmust open upward.

No; in the equation forthis graph, the numbermultiplied by x2 mustbe positive since thegraph opens upward.

No; the equation for thisgraph must have a positiveconstant since the graphcrosses the y-axis above 0.

082a-107a TE 02/2 MSMgr8 5/22/04 9:46 AM



10. Could this be the graph of the equation y � x2 � 1? Explain.

11. Graph A is the graph of y � x2.

a. Graph B is the graph of either y � 2x2 or y � �

x2

2

�.Which equation is correct?Explain how you know.

b. Graph C is the graph ofeither y � 3x2 or y � �

x3

2

�.Which equation is correct?Explain how you know.

12. Consider these three graphs.

a. Could Graph X be the graph of y � x2 � 1? Explain.

b. Could Graph Y be the graph of y � (x � 1)2? Explain.

c. Could Graph Z be the graph of y � �x2 � 1? Explain.

x

y

Graph Z

Graph YGraph X

x

yGra

ph C

Graph A

Graph B

x

yNo; the equation for thisgraph does not have aconstant of 1 since thegraph crosses the y-axisat 0.

11a. y � �x2

2�; Since Graph B

is wider than Graph A,the number multipliedby x2 must be lessthan 1.

11b. y � 3x2; Since GraphC is narrower thanGraph A, the numbermultiplied by x2

must be greater than 1.

12a. No; Graph X issymmetrical about avertical line left of they-axis. The graph ofy � x2 � 1 would besymmetrical about they-axis.

12b. No; Graph Y is sym-metrical about they-axis. The graph y � (x � 1)2 would be symmetrical aboutthe line x � �1.

12c. No; Graph Z has itsvertex to the left of(0, 1). The graphof y � �x2 � 1would have its vertexat (0, 1).

Exercise 11:Students may not see that y � �

x22� is equivalent to

y � �12�x2. If so, ask them to

substitute some values forx into each equation tosee whether the y valuesare the same. They shouldrecall that dividing by 2 isthe same as multiplying by�12�, so the arithmeticprocess, and not just theresults, should convincethem these are indeed thesame.

Exercise 12:This exercise is a goodassessment tool. Tocomplete it, studentsneed to understand theeffects of c and a iny � ax2 � bx � c andthe effects of h in y � (x � h)2.

AssessAssign and

082a-107a TE 02/2 MSMgr8 5/22/04 9:46 AM



13. Sports A place kicker on a football team attempted three fieldgoals during a game. All three were kicked from the opponent’s40-yard line, which is 50 yards from the goalpost. For a field goalto count, it must clear the crossbar, which is 10 feet high.

The football followed a different path through the air for each kick.These equations give the height of the kick in feet, h, for any dis-tance from the kicker in yards, d. Each kick was aimed directly atthe center of the goalpost.

Kick 1: h � 3.56d � 0.079d2

Kick 2: h � 1.4d � 0.0246d2

Kick 3: h � 2d � 0.033d2

a. For each kick, plot enough points to draw a smooth curve. Plot allthree graphs on the same axes and label them Kick 1, Kick 2, andKick 3. Put distance from the kicker, from 0 to 70 yards, on thehorizontal axis. Put height, from 0 to 50 feet, on the vertical axis.

b. Use your graphs to estimate the maximum height of each kick.

c. Use your graphs to estimate how many yards each kick traveledover the field before it struck the ground.

d. To make a field goal, the football must cross over the goalpost.That means it must be at least 10 feet high when it reaches thepost, which is 50 yards from the kicker.

Use your graphs to estimate whether any of the kicks could havescored a field goal. Explain your reasoning.

13a. See AdditionalAnswers.

13b. Kick 1: about 40 ft;Kick 2: about 20 ft;Kick 3: about 30 ft

13c. Kick 1: 45 yd;Kick 2: 57 yd;Kick 3: 61 yd

13d. Kick 1 did not reach the goalpost because the ball hit the ground after about45 yd. Kick 2 reached the goalpost, but passed under the crossbar at a heightof about 9 ft when d was 50. Kick 3 could have scored a field goal because itpassed over the crossbar at a height of about 18 ft when d was 50.

Exercise 13:Make sure you allow forenough time when youassign this problem, asstudents are asked tograph three quadraticequations and may nothave the use of a graph-ing calculator at home.

• Additional Answers on page A366

082a-107a TE 02/2 MSMgr8 5/22/04 9:47 AM



Tell whether each relationship is quadratic.

14. y � 3x � 5 15. y � �(x � 1)2

16. y � �5x

� 17. y � (x � 1)2

18. y � �(x � 1) 19. y � �x52�

20. y � (3x � 5)2 21. y � 7x3 � 3x2 � 2

22. y � x3 � 7x � 5 23. y � 2x

24. Consider these three equations.

y � x � 3 y � x2 � 3 y � x3 � 3

a. For each equation, draw a rough sketch showing the generalshape of the graph. Put all three graphs on one set of axes.


25. Graph A is the graph of y � x3.

a. Graph B is the graph of either y � �x2

3

� or y � 2x3. Which equationis correct? Explain.

b. Graph C is the graph of either y � �x3

3

� or y � 3x3. Which equationis correct? Explain.y � 3x3; the graph is narrower than the graph of y � x3.

x

y Graph C

Graph A

Graph B

nono

noyes

nono

yesno

yesno

24a.

24b. All three graphspass through (0, 3).The first graph is aline; the others arecurves. The secondgraph has a lowpoint at (0, 3). Thethird graph changesdirection at (0, 3).The second graphis symmetric aboutthe y-axis; the othersare not. In the firstand third graphs,the y values increasefrom left to right. Inthe second graph,the y valuesdecrease until (0, 3),and then increase.

25a. y � �x2

3�; the graph

is wider than thegraph of y � x3.

x

y

y � x � 3y � x2 � 3

y � x3 � 3

Exercise 16:You may want to ask ifstudents think the relation-ship is linear. (Of course,it’s not.) This exercise is agood preview for the nextlesson.

Exercise 24:Students should be able tograph the three relation-ships with relative ease.Encourage them to talkabout how these graphsrelate, respectively, to thegraphs of y � x, y � x2,and y � x3. (Adding 3 toeach expression translateseach graph up 3 units.)

AssessAssign and

082a-107a TE 02/2 MSMgr8 5/22/04 9:47 AM



26. The quadratic equations below are more complicated than those youworked with in Investigation 1. Just as with the simpler equations,you can make a table of values and plot a graph of the equation.

y � 3m2 � 2m � 7 p � n2 � n � 6 s � 2t2 � 3t � 1

Here is a table of values and a graph for y � 3m2 � 2m � 7.

m �3 �2.5 �2 �1 �0.5 0 0.5 1 2 2.5

y 28 20.75 15 8 6.75 7 8.75 12 23 30.75

a. Prepare a table of ordered pairs for p � n2 � n � 6. Plot thepoints on graph paper, and draw a smooth curve through them.

b. Prepare a table of ordered pairs for s � 2t2 � 3t � 1. Plot thepoints on graph paper, and draw a smooth curve through them.

c. How are the graphs for the three equations alike?

d. How do the three graphs differ? In particular, where are theirlowest points, and where are their lines of symmetry?

27. Each of these four graphs represents either y � x2 � 1 or y � 2x2 � 1.

a. Match each graph to its equation. Explain your reasoning.

b. How can graphs that look different have the same equation?

43210�1�2�3

35

30

25

20

15

10

5

�4

y

m

&ConnectExtend

26a. Tables will vary.

26b. Tables will vary.

26c. Possible answer: Allare parabolas thatcurve upward, andall have a verticalaxis of symmetrynear the y-axis.

26d. See AdditionalAnswers.

27. See AdditionalAnswers.

3

6

�1

�3

s

t

6

20

�10

�6

p

n

x

y

�2 2

5

4

3

2

1 x

y

�2 2

5

4

3

2

1 x

y

�2 �1 210�1 10�1 10

5

4

3

2

1 x

y

�2 �1 0 1 2

5

4

3

2

1

Graph A Graph B Graph C Graph D

Exercise 26:This exercise may takestudents some timebecause they need tocreate tables of orderedpairs for two quadraticfunctions and then plotthe points.

Additional Answers26d. Possible answer: The graphs differ in the location of their lowest points and lines of symmetry. y � 3m2 � 2m � 7: Lowest point is above

and to the left of the origin; line of symmetry is a vertical line to the left of the vertical axis. p � n2 � n � 6: Lowest point is below andslightly to the left of the origin; line of symmetry is a vertical line to the left of the vertical axis. s � 2t2 � 3t � 1: Lowest point is to the rightof and slightly below the origin, line of symmetry is a vertical line to the right of the vertical axis.

27a. Graphs A and B represent y � x2 � 1; Graphs C and D represent y � 2x2 � 1. The scales for the horizontal axes are the same forGraphs A and D, showing that Graph D is narrower than Graph A and therefore Graph D represents y � 2x2 � 1 and Graph A repre-sents y � x2 � 1. Similar reasoning shows that Graph C represents y � 2x2 � 1 while Graph B represents y � x2 � 1.

27b. The horizontal scales are different. When plotting the same equation on the different axes, one will be narrower than the other, even thoughthey represent the same equation.

082a-107a TE 02/2 MSMgr8 5/22/04 9:48 AM


28c. Tables will vary.

6

�4

�2

y

x


28. Challenge Consider the equation y � x2 � 4x.

a. Identify the values of a, b, and c in this quadratic equation.

b. Where does the graph of y � x2 � 4x cross the y-axis? How doesthis point relate to the value of c in the equation?

c. Graph y � x2 � 4x by making a table and plotting points. Makesure your graph shows both halves of the parabola.

d. Give the coordinates of the points where the graph crosses thex-axis.

e. Use the distributive property to write x2 � 4x as a product of twofactors.

f. How do the points where the graph crosses the x-axis relate tothis factored form?

29. Passengers in a hot air balloon can see greater and greater distancesas the balloon rises. The table shows data relating the height of a hotair balloon with the distance the passengers can see—the distance tothe horizon.*

Study the table to observe what happens to d as h increases by equalamounts.

x(x � 4)

(0, 0); (4, 0)

See below.Explain how thevalues of a and caffect the graphof the quadraticequationy � ax2 � c.

ownIn y o u r

words

28a. a � 1, b � �4,c � 0

28b. (0, 0); The value ofc is the y-coordinateof the point wherethe graph crossesthe y-axis.

*Adapted with permission from the Language of Functions and Graphs, p. 110. Shell Centre for Mathematical Education,University of Nottingham. Published Dec. 1985 by the Joint Matriculation Board, Manchester.

In May 1931, AugustePiccard of Switzerlandbecame the firstperson to reach thestratosphere when heballooned to almost52,000 feet. In October1934, Jeanette Piccardbecame the firstwoman to reach thestratosphere whenshe and her husband(Auguste’s twin brotherJean) ballooned toalmost 58,000 feet.

factsJust

t h e

The x-coordinate of each point makes one ofthe factors equal to 0.

Height Distance (meters), to Horizon

h (kilometers), d0 5

10 11

20 16

30 20

40 2350 25

Exercise 28:When you assign thisexercise, encouragestudents to be “smart”about the way they pickpoints for their tables. Youmay want to ask them tobegin with the values of xthat make y equal to 0 (0and 4). Then ask themwhat the x-coordinate ofthe vertex must be(halfway between, atx � 2). This kind ofnumerical analysis isextremely important forstudents to master. Theywill get more directexposure to this idea inlater chapters.

Exercise 29:This exercise is worthreviewing as a class, asit focuses in on the distinc-tions between linear andquadratic functions asthey relate to constantdifferences.

AssessAssign and

082a-107a TE 02/2 MSMgr8 6/4/04 1:46 AM

4


29b.

29c. Possible answer: Yes;the graph looks like itcould be half of aparabola.

50

25

0

d

h

Height (m)

Dis

tanc

e to

H

oriz

on (

km)


a. Below are three graphs. Which is most likely to fit the data in thetable? Explain.

b. Check your answer to Part a by sketching a graph that representsthe relationship between d and h.

c. Look at d in relation to h. Do you think these data could representa quadratic relationship? Explain.

30. Consider this pattern of cube figures.

a. Copy and complete the table for the pattern.

Stage, n 1 2 3 4

Cubes, C 0 4 18 48

b. How many cubes will beneeded to build Stage 5?

c. Write an equation forthe number of cubes inStage n. Explain howyou found your answer.

Stage 1 Stage 2 Stage 3 Stage 4

29a. The third graph; dincreases by smalleramounts as hincreases.

h

d

Height (m)

Dis

tanc

e to

H

oriz

on (

km)

10 20 30 400

h

d

Height (m)

Dis

tanc

e to

H

oriz

on (

km)

10 20 30 400

h

d

Height (m)

Dis

tanc

e to

H

oriz

on (

km)

10 20 30 400

In this graph, d increases In this graph, d increases In this graph, d increases by constant amounts as by greater amounts as by smaller amounts as

h increases. h increases. h increases.

On March 20, 1999,Bertrand Piccard(Auguste Piccard’sgrandson) and BrianJones became the firstaviators to circle Earthnonstop in a hot airballoon. Traveling the42,810 kilometers tookthem 19 days, 1 hour,49 minutes.

factsJust

t h e

30b. 100

C � n(n)(n � 1), or C � n2(n � 1),or C � n3 � n2; Possible explana-tions: Multiply length � width �height of each design: n � n � (n �1),which is equivalent to n3 � n2. Or, ifyou build a large cube with n3 unitcubes, and remove one layer(containing n2 unit cubes), you geta volume of n3 � n2.

Just the Facts:On July 2, 2002, SteveFossett became the firstsolo aviator to circle Earthin a hot air balloon.

082a-107a TE 02/2 MSMgr8 5/22/04 9:49 AM


31g.

60

70

�4

y

x

y � x2

y � 2x


31. Sometimes people confuse the quadratic relationship y � x2 withthe exponential relationship y � 2x.

a. Copy and complete the table of values for y � x2 and y � 2x.

x �4 �3 �2 �1 0 1 2 3 4 5 6

y � x2 16 9 4 1 0 1 4 9 16 25 36

y � 2x�116� �

18

� �14

� �12

� 1 2 4 8 16 32 64

b. For which values of x shown are x2 and 2x equal?

c. For which values of x shown is x2 greater than 2x?

d. For which values of x shown is 2x greater than x2?

e. Compare the way the values of y � x2 change as x increases by 1to the way the values of y � 2x change as x increases by 1.

f. How do you think x2 and 2x compare for values of x greaterthan 6?

g. Use the table of values to graph y � x2 and y � 2x on the sameset of axes.

Evaluate each expression for the given values.

32. h(g � hi) for h � 2, g � �1, and i � 5

33. ad � ab � bc for a � �3, b � �2, c � 3, and d � 2

34. (m � km)k for m � �4 and k � 3

Find the value of b in each equation.

35. 3b � 27

36. bb � 256

37. (�b)b � �27

Find the slope of the line through the given points.

38. (�2, 3) and (5, 8)

39. (0, �6) and (�8, 0)

40. (�3.5, 1.5) and (0.5, 2)

41. (�7, �2) and (�9, �2) 0

�18

�

��34

�

�57

�

3

4

3

512

23

�66

See below.

2x will be greater than x2.

0, 1, 5, 6

2, 431c. �4, �3, �2, �1, 331e. The values of x2

decrease untilx � 0, and thenincrease; theychange by �7, �5,�3, �1, 1, 3, andso on. The valuesof 2x double eachtime the x valueincreases by 1.

ReviewMixed

Exercise 31:Students may confusey � x2 and y � 2x

because they look sosimilar. Encouragestudents to talk about themeaning of each equation,looking at the table forexamples.

Quick Check

Informal AssessmentStudents should be able to:

✔ understand the effectof a, b, and c on thegraphs of parabolasin the form y � ax2 �

bx � c

✔ use quadratic equationsand graphs to analyzethe motion of objectsthrown into the air

✔ distinguish betweenquadratic relationshipsand other types ofrelationships, suchas cubics

Quick Quiz

1. Graph A is the graphof y � x2. Match theother two graphs withthe equations given:

�5 5

y

x

A

C

B

10

�10

y � (x � 1)2 Graph B

y � �x2 � 1 Graph C

• continued on next page

AssessAssign and

082a-107a TE 02/2 MSMgr8 5/22/04 9:50 AM


42. Consider this rectangle.

a. Write an expression for the rectangle’sarea.

b. Use your expression to write an equation stating that therectangle’s area is 108 square units.

c. Solve your equation to find the value of f.

43. Consider this rectangle.

a. Write an expression for the rectangle’sperimeter.

b. Use your expression to write an equation stating that therectangle’s perimeter is 39 units.

c. Solve your equation to find the value of f.

44. Rachel divided her fossil collection into three categories: plant fos-sils, insect fossils, and other animal fossils. She had eight more plantfossils than animal fossils. The number of insect fossils she has isthree less than four times the number of plant fossils she has.

a. If a represents the number of animal fossils Rachel has, write anexpression for the number of plant fossils she has.

b. Write an expression for the number of insect fossils Rachelhas.

c. Rachel has 41 more insect fossils than animal fossils. Usethis fact, and your expression from Part b, to write an

equation for this situation.

d. Solve your equation to find how many fossilsRachel has in each category.

4(a � 8) � 3

a � 8

10.5

9 � f � 9 � f, or 18 � 2f 9

f

12

9f � 108

9f 9

f

43b. 9 � f � 9 � f � 39,or 18 � 2f � 39

44c. 4(a � 8) � 3 � 41 � a or4(a � 8) � 3 � a � 41

44d. a � 4, so she had 4animal fossils, 12 plantfossils, and 45 insect fossils.


4. Explain how the graph of y � 2x3 is different from each of the following.

a. y � x3 The graph of y � 2x3 is thinner.

b. y � 2x3 � 1 The graph of y � 2x3 is 1 unit lower.

c. y � x2 Possible answer: The graph of y � 2x3 has a completelydifferent shape; it always goes up from left to right while y � x2

goes down and then up.

2. Table A represents therelationship y � x2.Using this table forcomparison, write therelationship representedby Table B. y � x2 � 2

Table A Table B

x y x y�3 9 �3 11�2 4 �2 6�1 1 �1 30 0 0 21 1 1 32 4 2 63 9 3 11

3. Consider the equationy � x2.

a. Will the graph ofy � ��

13�x2 � 2 be

thinner or wider thanthe graph of y � x2?wider

b. Will the graph ofy � ��

13�x2 � 2 open

up or down? down

c. Will the graph ofy � �

13�x2 � 2 have

its vertex on they-axis? How do youknow? Yes; there isno x term (b � 0).

d. What is the equationof the graph that isy � �

13�x2 � 2 moved

2 units down?y � �3

1�x2

082a-107a TE 02/2 MSMgr8 5/22/04 9:50 AM

107a C H A P T E R 2 Quadratic and Inverse Relationships

Teacher Notes

082a-107a TE 02/2 MSMgr8 5/22/04 9:51 AM

families of quadratics - glencoe.com

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