Algebra II

Note workbook

Chapter 2

Name_______________________

Algebra II: 2-1 Relations and Functions

The table shows the average lifetime and maximum lifetime for some animals. This data can be written as ________________________________________. The ordered pairs of the data are ___________________ _______________________________. The first number in each ordered pair is the ___________________ and the second number is the ____________________________________________. We represent ordered pairs by ______________________________. When we graph, we graph on a coordinate system with __________________________. The two axis coordinate system is called the _________________________________________. The horizontal axis is labeled the _________________ and the vertical axis is labeled the __________________. The point at Which the two axes meet is called the _________________ and its coordinates are ______________. The four regions divided by the coordinate axes are called ___________________________. We ALWAYS write ordered pairs in the form ____________. Ex. 1: Label all of the important parts of the coordinate axis below.

A ___________________________ is a set of ordered pairs. The ____________________________ is the set

of all first coordinates (___________________________) and the ____________________ is the set of all

second coordinates (______________________________).

Ex. 2: Consider the set: {(-1, 3), (2,4) , (3, 1), (8, 6)}.

a) Find the domain of the set. b) Find the range of the set.

A _______________________ is a special type of relation in which each element on the domain is paired with

______________________ one element of the range. We often draw __________________ as a way to

represent functions.

A special function is a _________________ function. In this kind of function each element of the range is

paired with exactly one element of the domain.

Ex. 3: State the domain and range of the relation shown in the graph. Is the graph a relation?

Ex. 4: Is the set {(9, 3), (9, -3), (4, 2), (4, -2)} a function? Construct a mapping of this relation.

If a relation is graphed, it is very easy to tell if a graph is a function. We use the _________________________.

Ex. 5: Are the following relations functions?

a) b)

c) d)

Ex. 6: The table shows the average fuel efficiency in miles per gallon for light trucks for several years. Graph

this information and determine whether it represents a function.

So far, we have learned three ways that relations and functions can be represented:

1) ___________________________

2) ___________________________

3) ___________________________

The fourth way to represent relations and functions is by writing an _______________________. YOU NEED

TO BE ABLE TO MOVE BACK AND FORTH BETWEEN ALL FOUR REPRESENTATIONS OF

FUNCTIONS!!!!!!!!!!!!!!!!!!

The __________________________ of an equation in x and y are the set of ordered pairs ________ that make

the equation ___________________.

Consider the equation � � 2� � 6.

What can x be? _____________________. So, we say that the domain is ______________________.

Ex. 7: Graph the relation by represented � � 2� � 1.

Ex. 7 con’t.: b) Find the domain and range of � � 2� � 1.

b) Is this relation a function?

When an equation represents a function, the variable, usually x, whose values make up the domain is called the

_______________________________________________________. The other variable, usually y, is called the

_______________________________________________________ because its values depend on x.

When equations are represented by functions, we usually write the equation in

_______________________________________________. For example, we can write the equation y = 2x +1

as f (x) = 2x +1. The symbol f (x) replaces the y. The f is just the ___________________ of the function. It is

NOT a variable that is multiplied by x. Suppose you want to find the value in the range that corresponds to the

element 4 in the domain of the function. This is written as f (4). The value f (4) is found by substituting 4 for

each x in the equation. So, f (4)=________________________________________________________.

Ex. 8: Given �� � �� � 2 and �� � 0.5�� � 5� � 3.5, find each value.

a) �3� b) 2.8� c) 3��

HW: Day One

1)

2) Graph each relation or equation and find the domain and range. Then determine whether the relation or

equation is a function.

a. �2,1�, �3,0�, 1,5�� b. �4,5�, 6,5�, 3,5��

c. � � �5� d. � � 3�

3) For question 3, use the table that shows a company’s stock price in recent years.

a. Write a relation to represent the data.

b. Graph the relation and tell if it is a function.

c. Identify the domain and range.

a) b) c)

d) e) f)

HW: Day Two:

1) Find each value if �� � 3� � 5 and �� � �� � �.

a. �3� b. 3� c. �

��

d. �

�� e. �� f. 5��

2) Find the value of �� � �3� � 2 when � � 2.

3) What is 4� if �� � �� � 5?

4) If �� � 2� � 5, then 0� �

a. 0 b. – 5 c. – 3 d. �

�

5) If �� � ��, then � � 1� �

a. 1 b. �� � 1 c. �� � 2� � 1 d. �� � �

Algebra II: 2-2 Linear Equations

Definition: A ____________________________ is an equation that has no operations other than

__________________, ________________________, and multiplication of a variable by a constant. The

variables MAY NOT be _______________________ together or appear in a __________________________.

All variables have exponents of _________.

*******KEY IDEA: The graph of a linear equations is ALWAYS a ________________________.

Linear Equations Not Linear Equations

5� � 3� � 7 7� 4�� � �8

� � 9 � � √� 5

6� � �3� � 15 � �� � 1

� �12� � �

1�

Def: A _________________________________________ is a function whose ordered pairs satisfy a linear

equation. Any linear function can be written in the form _________________________________, where m and

b are real numbers.

Ex. 1: State whether the function is a linear function.

a) ���� � 10 � 5�

b) ���� � �� � 5

c) ���, �� � 2��

Ex. 2: The linear function ���� � 1.8� 32 can be used to find the number of degrees Fahrenheit, f, that are

equivalent to a given number of degrees Celsius, C.

a) On the Celsius scale, normal body temperature is 37oC. What is the normal body of temperature in

degrees Fahrenheit?

b) There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit

degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree?

Any linear equation can be written in _________________________. The standard form of a linear equation is

________________________________________, where A, B, and C are integers whose greatest common

factor is 1.

Ex. 3: Write each equation in standard form. Identify A, B, and C.

a. � � 3� � 9

b. � ��� � 2� � 1

c. 8� � 6� 4 � 0

Special points on a line. One of the quickest ways to graph a line is to identify where the line crosses the

____________ and the _____________. The point where the line crosses the x-axis is called the ____________

and the point where the line crosses the y-axis is called the _____________________________.

To find the x- and y- intercepts:

To find the x-intercept, let y = 0. To find the y-intercept, let x = 0.

Ex. 4: Find the x-intercept and the y-intercept of the graph of 3� � 4� 12 � 0. Then graph the equation.

HW: 1) State whether the function is linear.

a. �� �� � 4 b. ���� � 1.1 � 2� c. � � � 5

d. ! 3� � 5 e. � "� � 4 f. ���� � 2�� � 4�� 5

2) Which of the following equations is NOT linear?

a. � 9� � 7 b. �� 5� � 0 c. � � 3� � 1 d. � � �2

3) Which of the following equations is a linear function?

a. � 3 � ! b. �� � 1 c. � � 3 � � d. �� � 9

4) When a sound travels through water, the distance � in meters that the sound travels in � seconds is given by

the equation � � 1440�.

a. How far does a sound travel underwater in 5 seconds?

b. In air, the equation is � � 343�. Does sound travel faster in air or water? Explain.

5) Write each equation in standard form.

a. � � �3� 4 b. � � 12� c. � � 4� � 5

d. 5� � 10� � 25 e. ��

�� � 6 f. 0.5� � 3

6) Find the � � and � �intercepts of the graph of each equation and then graph the equation.

a. 5� 3� � 15

b. 2� � 6� � 12

c. 2� 5� � 10 � 0

d. � � 4� � 2

e. � � �2 f. � � 8

7. Suppose the temperature T (oC) below the Earth’s surface is given by #�$� � 35$ 20, where d is the depth

(km).

a. Find the temperature at a depth of 2 kilometers.

b. Find the depth if the temperature is 160oC.

c. Graph the linear function.

8. The Jackson Band Boosters sell beverages for $1.75 and candy for $1.50 at home games. Their goal is to

have total sales of $525 for each game.

a. Write an equation that is a model for the different numbers of beverages and candy that can be sold to meet

the goal.

b. Graph your equation.

c. Does this equation represent a function?

d. If they sell 100 beverages and 200 pieces of candy, will the Band

Boosters meet their goal?

Algebra II: 2-3 Slope

Definition: The ___________________ of a line is the ratio of the change in y-coordinates to the corresponding

change in x-coordinates.

Suppose a line passes through the points ���, ��� and ���, ���. Then, slope = ��� �� ���������� �

��� �� ���������� � �

�����

�����.

Ex. 1: Find the slope of the line that passes through (1, 3) and (-2, -3). Then graph the line.

Ex. 2: Graph the line passing through (1, -3) with a slope of ��

�.

The slope of a line tells the direction in which the line _____________________ or _________________.

In real life and in science classes, slope is often referred to as the _______________________________. It

measures how much one quantity changes on average relative to the change in another quantity, often time.

Examples of common rates of change include: ________________________________________________.

Ex. 3:

Parallel and Perpendicular Lines:

Using a graphing calculator, draw the graphs of � � 3� � 2, � � 3� � 2, and � � 3� � 5 on the same

coordinate plane.

What do you notice about all 3 of these graphs?

Can you make a conjecture based on what you see?

Ex. 4: If line a is parallel to the line 3� � 4� � 8, what is the slope of line a?

Ex. 5: Graph the line through �1, �2� that is parallel to the line with equation � � � � �2.

We say that the slopes are ______________________________________________ of each other. When you

multiply the slopes of two perpendicular lines, the product is always _____________.

Ex. 6: Suppose line l is perpendicular to the line given by 2� � � � 4. What is the slope of line l?

Ex. 7: Graph the line through (2, 1) that is perpendicular to the line with equation 2� � 3� � 3.

HW: Day One

1) Find the slope of the line that passes through each pair of points.

a. �6,1�, �8, �4� b. ��6, �5�, �4,1�

c. �7,8�, �1,8� d. �4, �1.5�, �4,4.5�

2) Determine the value of ( so that the line through �6, (� and �9,2� has slope �

�.

3) Graph the line passing through the given point with the given slope.

a. �2,6�, * ��

� b. ��3, �1�, * � �

�

+

c. �3, �4�, * � 2 d. �1,2�* � �3

4) Refer to the graph that shows the number of CD’s and cassette tapes shipped by manufacturers to

retailers in recent years.

a. Find the average rate of change of the number of CD’s

shipped from 1991 to 2000.

b. Find the average rate of change of the number of cassette tapes shipped from 1991 to 2000.

c. Interpret the sign of your answer to part b).

5) Mr. And Mrs. Wellman are taking their daughter to college. The table shows

their distance from home after various amounts of time.

a. Find the average rate of change of their distance from home between 1

and 3 hours after leaving home.

b. Find the average rate of change of their distance from home between 0 and 5 hours after leaving

home.

c. C. What is another word for rate of change in this situation?

HW: Day Two:

1) Graph the line that satisfies each set of conditions.

a. Passes through ��2,2�, parallel to a line whose slope is �1.

b. Passes through ��4,1�, perpendicular to a line whose slope is ��

�.

c. Passes through �3,3�, perpendicular to the graph of � � 3.

d. Passes through �2, �1�, parallel to graph of 2� � 3� � 6.

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Date: ______________________

Section 2 – 4: Writing Linear Equations

Forms of Equations Consider the following graph. The line passes through ________ and ________. Notice that ______ is the y-intercept of _______. You can use these two points to find the slope of _______. Find the slope: Now solve your new equation for y. This form is called the ___________________________.

Example #1: Write an Equation Given Slope and a Point

Write an equation in slope-intercept form for the line that has a slope of 5

3− and passes

through (5, - 2).

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Point-slope form: use this form to find an equation of a line when you are given the ___________________ of two ____________ on a line

Example #2: Write an Equation Given Two Points Write an equation of the line through (2, - 3) and ( - 3, 7). When changes in real-world situations occur at a __________ ________, a linear equation can be used as a ___________ for describing the situation. Example #3: Write an Equation for a Real-World Situation Sales – As a part-time salesperson, Jean Stock is paid a daily salary plus commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. a) Write a linear equation to model this situation. b) What are Ms. Stock’s daily salary and commission rate? c) How much would Jean make in a day if her sales were $500?

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Parallel and Perpendicular Lines The ___________________ and ___________________ forms can be used to find equations of lines that are ________________ or ___________________ to given lines. Example #4: Write an Equation of a Perpendicular Line Write an equation for the line that passes through (3, - 2) and is perpendicular to the lien whose equation is 15 +−= xy .

HW:

1) State the slope and y-intercept of the graph of each equation.

a. � � � �� � � 4 b. � � �

� � c.2� � 4� � 10 d. 3� � 5� � 30 � 0 e. � � 7 f. �� � � � �

2) Write an equation in slope-intercept form for each graph.

a. b.

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c. d.

3) Write an equation in slope-intercept form for the line that satisfies each set of conditions. a. slope 3, passes through �0, �6� b. slope 0.25, passes through �0,4�

c. slope � �� , passes through �1,3� d. slope

��, passes through ��5,1�

e. x-intercept �4, y-intercept 4 f. passes through �2, �5�, perpendicular to � � �� � � 7

4)

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Date: ______________________

Section 2 – 5: Modeling Real-World Data: Using Scatter Plots

Real Numbers Data with two variables, is called ____________ ________. A set of bivariate data graphed as ordered pairs in a coordinate plane is called a _____________ ______. A scatter plot can show whether there is a ___________________ between the data. Example #1: Draw a Scatter Plot Education – The table shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data. Prediction Equations When you find a line that closely approximates a set of _______, you are finding a __________________ for the data. An equation of such a line is often called a _____________________ because it can be used to predict one of the variables given the other variable. To find a line of fit and a prediction equation for a set of data, select ________________ that appear to represent the data well. This is a matter of personal judgment, so your line and prediction equation may be different from someone else’s.

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Example #2: Find and Use a Prediction Equation Education – Refer to the data in Example #1. a) Draw a line of fit for the data. How well does the line fit the data? b) Find a prediction equation. What do the slope and y-intercept indicate? c) Predict the percent in 2010. d) How accurate is the prediction?

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HW: 1)

2) Whether you are climbing a mountain or flying in an

airplane, the higher you go, the colder the air gets. The table shows the temperature in the atmosphere at various altitudes. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use your prediction equation to predict the missing value.

3) As more channels have been added, cable television has become attractive to more viewers. The table shows the number of U.S. households with cable service in recent years.

a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation.

c. Use your prediction equation to predict the missing value.

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Date: ______________________

Section 2 – 6: Special Functions

Step Functions, Constant Functions, and the Identity Function The cost of postage to mail a letter is a _____________ of the weight of the letter. But the function is not __________. It is a special function called a ________ ___________. The graph of a step function is not _________. It consists of line segments or rays. The ___________ ___________ _____________, written _______________, is an example of a step function. The symbol _______ means the greatest integer less than or equal to x. For example, _________________ and _______________ because ________________. Study the table

and graph below. Example #1: Step Function Psychology – One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.

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You’ve learned that the slope-intercept form of a linear function is _____________, or in functional notation, ________________. When m=0, the value of the function is ___________ for every x value. So, f(x) = b is called a ______________ ____________. The function f(x) = 0 is called the __________ ____________. Example #2: Constant Function

Graph 3)( −=xg . First make a table of values.

Another special case of slope-intercept form is ___________, __________. This is the function _____________. The graph is the line through the ____________ with slope 1. Since the function does not change the input value, ______________ is called the ________________ _________________. Absolute Value and Piecewise Functions – Another special function is the _____________ __________ ______________, ______________.

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The absolute value function can be written as _________________________. A function that is written using two or more ___________________ is called a ________________ ________________. Recall that a family of graphs is a group of graphs that displays one or more similar _____________________. The parent graph of most absolute value functions is ____________. Example #3: Absolute Value Functions

Graph 3)( −= xxf and 2)( += xxg on the same coordinate plane. Determine the

similarities and differences in the two graphs.

To graph other piecewise functions, examine the ___________________ in the definition of the function to determine how much of each piece to include. Example #4: Piecewise Function

Graph

>−≤−

=3 if 1

3 if 1)(

x

xxxf .

Identify the domain and range.

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Example #5: Identify Functions Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function. a) b)

HW:

1) a. b. c. d. e. f.

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2) Bluffton High School chartered buses so the student body could attend the girls’ basketball state tournament games. Each bus held a maximum of 60 students. Draw a graph of a step function that shows the relationship between the number of students x who went to the game and the number of buses y that were needed.

3) Graph each function. Identify the domain and range. a. ���� � ��� � 3�� b. ���� � ��� � 2��

c. ���� � |�| � 3 d. ���� � |�| � 4

e. ���� � ���, � � 32, � � 3

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Date: ______________________

Section 2 – 7: Graphing Inequalities Graph Linear Inequalities A linear inequality resembles a linear equation, but with an inequality symbol instead of an ________ ______________. For example, _________________ is a linear inequality and _________________ is the related linear equation. The graph of the inequality __________________ is the ____________ region. Every point in the shaded region satisfies the inequality. The graph of _______________ is the ________________ of the region. It is drawn as __________ _______ to show that points on the line satisfy the inequality. If the inequality symbol were < or >, then points on the boundary would not satisfy the inequality, so the boundary would be drawn as a ____________ _________ You can graph an inequality by following these steps. Step 1 – Determine whether the boundary should be _____________ or ____________. Graph the boundary Step 2 – Choose a ___________ not on the boundary and test the inequality. Step 3 – If a ________ inequality results, shade the region containing your test point. If a ________ inequality results, shade the other region. Example #1: Dashed Boundary Graph 42 <− yx .

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Example #2: Solid Boundary Business – A mail-order company is hiring temporary employees to help in their packing and shipping departments during their peak season. a) Write an inequality to describe the number of employees that can be assigned to each department if the company has 20 temporary employees available. b) Graph the inequality.

c) Can the company assign 8 employees to packing and 10 employees to shipping?

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Graph Absolute Value Inequalities Graphing absolute value inequalities is similar to graphing ____________ inequalities. The inequality symbol determines whether the boundary is ___________ or __________, and you can test a point to determine which region to ______________. Example #3: Absolute Value Inequality

Graph 2−≥ xy .

HW: Graph each inequality

1) � � � � �5 2) 3 ! � � 3� 3) � � 6� � 2

4) � � |�| 5� � � |4�| 6� � � |�| " 3