Name:_____________________________ Period: ___________________

Objective 127a ______________________________________________________________________________

The Cartesian Coordinate System:

KEY VOCABULARY VOCABULARY/NOTATION Definition Picture

X-AXIS

Y-AXIS

ARROWS ON ENDS OF AXIS

The point where the two lines intersect

The four different regions of the graph

1

Objective 127b ______________________________________________________________________________

ESSENTIAL VOCABULARY

Ordered Pair _____________________________________________________________________________

( X , Y)

The _____________________ tells us exactly how far left or right to move from the ____________________________________.

The ______________________ tells us exactly how far up or down we need to move from the ____________________________________.

****TO REMEMBER ( x , y ) :

1. How many places do we need to move horizontally and in what direction for point D?

2. How many spaces did you need to move and in what direction for point D?

So D is _________right and _________down.

D went ________then___________,

So his ordered pair is (_____,______)

2

C

Point x-coordinate y-coordinate Orderedpair

Quadrant Signs in Quadrant

A

B

C

E

F (0, -3)

G (-4, -5)

Write the ordered pairs for points A, B, C, and D. Label points F and G. For all, name each quadrant in which each point is located. Then, state the sign of the x-coordinates and the y-coordinates in that particular quadrant.

YOU TRY!

1. LABEL each axis on the coordinate plane to the right.2. Plot AND LABEL each point on the coordinate plane to the right.

a. (3, 1) b. (-2, 0) c. C(2, -5)

d. (-5, 5) e. (0, -3) f. (0, 4)

3. State which quadrant each coordinate pair will be in: (-1, -1) ____________________quadrant.

(5, -2) ____________________quadrant.

(-, +) ____________________quadrant.

(2, 2) ____________________quadrant.

( -, -) ____________________quadrant.

(0, 6) _____________________ quadrant.

4. Label and plot the following points on the graph below. 5. Identify the coordinates of the given points on the graph.

A: (4, 5) B: (0,-3) A: B:

C: (4, 0) D: (-3, 4) C: D:

E: (-4, -2) F: (3, -3) E: F:

3

A

D

C

B

ADDITIONAL PRACTICE 1) Mr. Erwin started at the point (0, 5) on the coordinate grid. He went up three and to the right four. What is the

new point and what quadrant is it located in?

2) Indra started at the point (-4, -2). She moved three spaces up and five spaces to the right. What is the new point and what quadrant is it located in?

3) The point (1, 2) , is located on a map. The point moved located three blocks west and two blocks north. What is the new location of the point?

4) Ms. Stoecklin’s home is located at the point (2, -3) on the coordinate grid. The grocery store is located four blocks east and three blocks south. At what point on the coordinate grid is the grocery store located?

5) A local park in the shape of a square was built using the grid from a coordinate plane. The designers placed the center of the park on the origin. If the northeast corner of the park is located 4 blocks north and 5 blocks east,

4

what are the coordinate pairs of the northwest, southeast and southwest corners of the park? (HINT: Draw a picture!)

PUTTING IT TOGETHER 1. Given the graph to the right, which is located in the standard (x, y) plane,

which of the following points is located at ( 3, -3) a. Ab. Bc. Cd. D

2. Ms. Stoecklin graphed her house on the standard coordinate (x, y) plane and found that her house was at the point (3, 4). Work is located 4 blocks south and 2 blocks west. What point is work located at?

3. ABC is a triangle. If the coordinates of point C are 2 blocks south and 4 blocks west of the origin, what are the coordinates of point A and B?

Objective 128a ______________________________________________________________________________Holt McDougal Algebra 1 (Lesson 3-2) and 3-2 Lab

Glenco McGraw Hill (Lesson 4-3 then 4-6 example 1)

5

A

B

C

D

A

BC

6

Graphing Equations on Calculator.

Lesson 4-6 of Glenco—Examples 2, 3, 4, 5

Lesson 3-3 of Holt McDougal

7

Objective 128b-_______________________________________________________________

Functional Notation: _______________________________________________________________

__________________________________________________________________________________________________

___________ _____________________ y = 3x – 8 f(x) = 3x – 8

g(x) = 3x – 8 h(x) = 3x – 8

WRITE THE FOLLOWING IN FUNCTION NOTATION

1) y = 5x + 12 2) h=9x+9 3) b=3x+2x-8

Practice putting equations into function notation

1. y = 12 + 2x – 8 2. j = 4.5x + 31 3. y = 32 – 5d 4. c = 4w – 7w + 8

Write 5 equations and then put them into function notation (MUST USE DIFFERENT VARIABLES)

1) 2) 3)

8

4) 5)

Evaluating Linear Function Values

EXAMPLE 1:

If f(x) = 2x + 5, find each value

i. f(-3)

ii. f(1) + 4

iii. f(x + 4)

9

Evaluating Nonlinear Function Values

EXAMPLE 2:

If h(z) = z2 + 3z + 4, find each value.i. h(-2)

ii. h(3a)

iii. 2[h(n)]

Example 3) If (t )=(−4+2 t)−(t ) , then what is u(−4) equal to?

10

Putting It all Together

1) If

2) Give than

3) If h ( x )=(3 x2−10 ) (3x+2 ) ,than h (2 )=¿

11

Objective 129a- _________________________________________________________________________________

ESSENTIAL VOCABULARY

SLOPE ( ):

Rates of Change( ) Definition/ Math Picture

Positive

Negative

Zero

Undefined

STEEPNESS OF SLOPE

12

Looking at the picture of the mountain…

1) Is the slope of the right side of the mountain positive or negative?

2) Is the slope of the left side of the mountain positive or negative?

3) Which side of the mountain is steeper?

1)Which of the following has the smallest positive slope?

2)Which of the following has the smallest negative slope?

3)Which of the following has a largest positive slope?

4)Which of the following has the largest negative slope?

5) Draw a line with a slope of zero. Explain how you remember that this line has a slope of zero.

6) Draw a line with a slope that is undefined. Explain how you remember that this line has a slope that is undefined.

13

AB

C

D

E

The larger the________________________ for the rate of change, regardless of the sign, the…

The smaller the________________________ for the rate of change, regardless of the sign, the…

(x1,y1)

(x2,y2)

(y2 – y1)

(x2 – x1)

REPRESENTATION OF SLOPE (Rate of Change) IN ALGEBRA

What is slope?

Find the slope of the line connecting the points (-4, 1) and (4, -5) by graphing.

14

FINDING SLOPE, OR RATE OF CHANGE, FROM A GRAPH

15

16

FINDING RATE OF CHANGE WHEN GIVEN TWO ORDERED PAIRS

Example 1) Find the slope of the line that passes through (-1, 2) and (3, -4)

SLOPE=

Example 2) Find the slope of the line.

SLOPE=

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Example 3) Plot the points (-1, 2), (3, 2) on the graph.

SLOPE=

****SMART CUT: LINES WITH A SLOPE OF ZERO WILL HAVE…..

Example 4) Plot the points (1,3), (1, -2) on the graph

SLOPE=

****SMART CUT: LINES WITH AN UNDEFINED SLOPE WILL HAVE…..

YOU TRY!Find the slope of the line that passes through Find the slope of the line that passes through (-4, 5) and (1, 5) (-4, -1) and (-4, 2)

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CRITICAL THINKING

1) Given that slope of a line is 2, what can you conclude about the rise of a line if the run is 2?

2) Given that the slope of a line is less than 1, what can you conclude about the run of a line if the rise is 6?

3. Mrs. Moran set up a slide in her backyard. She stacked up 10 bricks and then laid a slide on top. Which of the following describes the relationship between the rise and the run if the slope of the slide was greater than 1?

a) The rise is greater than the run

b) The rise is less than the run

c) The slope of the line is zero?

d) The slope of the line is undefined

k) The rise and the run are the same

4. Mr. Erwin drew plans for his new shed. When constructing the roof, he drew a roof height of 5 feet and made the slope of the roof less than one. What can you conclude about the length of the roof?

a) The rise is less than the run

b) The rise is greater than the run

c) The rise and the run are the same

d) The slope of the line is undefined

k) The slope of the line is zero?

19

Objective 129b- _________________________________________________________________________________

Example 1) Find the slope of the line that passes through ( , ) and ( , ).

Example 2) Find the slope of the line that passes through the point ( , ) and ( , ).

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Example 3) Find the slope of the line that passes through the point ( , ) and ( , ).

SLOPE=

Example 4) What is the rate of change for the function in the table? Is it a linear rate of change? Justify your answer.

Example 5) Which function has the greatest rate of change? Justify your answer.

A) . B) (-2,11) (-1,8) (1,2)

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X Y

-6 -6

0 -1

6 4

12 9

X Y

-2 -4

0 2

1 5

3 11

INTERCEPTS

y-intercept

The y-intercept is __________________________________________________________________________________

EXAMPLE 1: Graph the ordered pairs (0,5), (0,1), and (0,-3). What do you notice?

SMART CUT: The x value in an ordered pair at the y-intercept is ALWAYS=____________

Write the ordered pair.

1) The y-intercept is 4 ___________ 3) The y-intercept is -2 ____________

2) The line crosses the y-axis at 1 ________ 4) The line crosses the y-axis at -6 ________

x-intercept

x-intercept is ___________________________________________________________________________________________

EXAMPLE 2: Graph the ordered pairs (4,0), (1,0), and (-2,0).

What do you notice?

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SMART CUT: The Y value in an ordered pair at the X-intercept

is ALWAYS=____________

Write the ordered pair.

(1) The x-int is 5 ___________

(2) The x-int is -3 ____________

(3) The line crosses the x-axis at 0 ________

(4) The line crosses the x-axis at -1 ________

Practice 1) What is the slope of a line that passes through point (-2,1) and has a y-intercept of 3?

Background Brainstorm… what do you know about the coordinates of y-intercepts?

Practice 2) What is the slope of a line that passes through point (1,-3) and crosses the x-axis at 5?

Practice 3) What is the slope of a line that passes through the origin and point (2,4)?

Practice 4) What is the slope of a line that has an x-intercept of -3 and crosses the y-axis at 5?

Practice 5) Write your own problem similar to the ones above and solve it.

23

Think It Out:

1) If the slope of a linear function is positive, but less than one, what all do you know about the slope?

2) If the slope of a linear function is positive and more than one, what all do you know about the slope?

3) If the slope of a linear function is undefined, what all do you know about it?

4) If the slope of a linear function is zero, explain what you know about the slope?

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Objective 130a: __________________________________________________________________________

ESSENTIAL VOCABULARY

WORD Definition PICTURE

LINEAR

SLOPE

Y-INTERCEPT

SLOPE INTERCEPT FORM

y = mx + b

Y= M= B= X and Y

Find the slope of a line given the following points that are on the line.

1) Passes through (3,5) and (4,-3) 2) Passes through (1,3) and (-2,-3)

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For each graph, determine whether the slope is positive or negative, and explain how you know

1. Which ordered pair is on the x-axis?

a. (0,-2)

b. (-2, 0)

c. (-2, -2)

2. Which ordered pair is on the y-axis?

a. (0,-2)

b. (-2, 0)

c. (-2, -2)

CFU: Which formula correctly shows the slope-intercept form?

A. a2 + b2 = c2

B. y2 = mx2 + b

C. y + mx = b

D. y = mx + b

26

CFU: What two things do you need to write an equation of a line in form y=mx+b?

1)_______________________________ and 2)_________________________________

KEY POINTS: The slope must be written as a __________________________, and is typically not

written as a _________________________________________ or a _______________________________________.

If the y—intercept is not an integer, it is best to write it as a ___________________________.

Writing the equation of a line in slope intercept form given the slope and the y-intercept

Example 1) Write the equation of the line given a slope of 3 and a y-intercept of 4

Example 2) Write the equation of a line given m=1/2 and b=−5

Example 3) Write the equation of a line in slope-intercept form given b=-1 and m=0

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Example 4) Given that slope is -1 and y-intercept is 0, what is the equation of the line in slope intercept form?

Example 5) Write the equation of a line in slope-intercept form with a slope of 0.75 and a y-intercept of 52

YOU TRY: Write the following in slope-intercept form.

1) Slope: 5 , y-intercept:-3 2) m= -2, y-intercept: 7 3) Slope: -6, y-intercept: -2

4) m=3 12 , b= 1 5) m= 3, b=

65 6) Slope: -4, y-intercept: 0

7) Slope: 1, y-intercept: -4 8) b= 8 , m= 0 9) Slope= -1, y-intercept=0

Graph each of the y-intercepts from problems 1-8 above. Label your y-intercepts with the problem number.

Questions 1, 2, 3 Questions 4, 5, 6 Questions 7, 8, 9

Directions: Write the slope-intercept equation given the following information

1) Slope=2, y-intercept= -3 ___________________________

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2) Slope=0, y-intercept= -5 ___________________________

3) Y-intercept= 112 , slope= -0.2 ___________________________

4) Y-intercept=0, Slope= 122 ___________________________

5) Slope= −13 , y-intercept=1 ___________________________

Critical Thinking 1: Graph A has the equation y=5 x+b, and Graph B has the equation y=cx+b. Graph A has a steeper positive slope than graph B. What must be true about the relationship between the equations for graphs A and B.

a) b=b b) 5<cc) 5>c d) a=c e) Not enough information

Justify your answer:

Critical Thinking 2: In the graph below, graph A has the equation y=ax+b, and Graph B has the equation y=cx+b. Which of the following must be true? Explain how you made your decision.

a) b=b b) a<cc) b>c d) a=c e) Not enough information

29

A

B

PUTTING IT TOGETHER

1) A line with equation y=mx+b has a smaller y-intercept than a line with equation y = 3x -6. Which of the following statements must be true?

a. b > -6 b. b < -6 c. b > 3 d. m > -6 e. m > 3

2) A line with equation y=mx+b has a larger positive slope than a line with equation y = 1/5x +6. Which of the following statements must be true?

f. b > 6 g. b < 1/5h. b > 5i. m > 1/5

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Objective 130b________________________________________________________________________

What two things do you need to write an equation of a line? 1)_________________________ 2)_____________________________

Example 1: Slope of −34 and through (8,2) Example 2: (-1, 3), m = 1/5

Example 3: (4, -2), m = 5 Example 4: (3, 2) , b=3

Practice 1: (1, 2), m = -3 Practice 2: (-1, 2), m = 1/3

Practice 3: (1, -2), m = 1/2 Practice 4: (2, 2) , b=1

31

Practice 5: Through (0, -2) with a slope of 2 Practice 6: (2, 1) , b=-2

32

Objective 130c-__________________________________________________________________________

What do we need to write the equation of a line?

SMART CUT FOR B:

WRITE THE EQUATION OF A LINE GIVEN TWO POINTS

EXAMPLE 1: Write the equation of the line containing points (3, -1) and (-2, -6)

EXAMPLE 2: Write the equation of the line containing points (1, 2) and (2, -4)

33

EXAMPLE 3: Write the equation of the line containing points (-3, 1) and (0, -4)

EXAMPLE 4: Write the equation of the line that passes through the origin and (3,2)

EXAMPLE 5: Write the equation of the line that has an x-intercept of -4 and a y-intercept of 6.

EXAMPLE 6: Write the equation of the line containing with the input/output table below.

X Y-4 60 1

34

Objective 130d-__________________________________________________________________________

What two things do you need to write the equation of a line?

Steps for Writing the Slope-Intercept Equation from a Graph

1. Find the y-intercept and write b=¿ ( , )2. Determine the slope and write m=¿

a. Determine the sign of the slopeb. Make two points on a graph where the three lines perfectly intersectc. Draw a line up from your lowest point, and from your nearest point above it draw a line over to form

a right triangled. Label your triangle for the number of units up and the number of units over the line ise. Write your rise (up and down number) over your run (left and write number)

3. Write the equation y=m x+b using your numbers from steps one and two above4. **If the line crosses through the origin, we don’t need to write the b because it is _______________

KP:

Example 1 Example 2

35

1)

2)

3)

1)

2)

3)

Writing Equations of Vertical and Horizontal Lines

Directions: Write three ordered pairs for each graph below.

What do you notice about your coordinates for the lines above?

The line always crosses at: The line always crosses at:

Draw your own vertical and horizontal lines, then write the equation for each line.

36

If we are taking a multiple choice test, we can use shortcuts like process of elimination to eliminate some of the choices

Steps for Using Smartcuts

1. Examine the y-intercept, pay special attention to whether it is positive or negative2. Look at the sign of the slope and eliminate the choices that are not the same sign3. Look at the steepness of the slope and eliminate choices that are not the same4. Check your answer by doing the longer method to make sure your answer is correct

Example 1 Example 2

Explain how you quickly know each of the other answer choices is incorrect.

37

A)

B)

C)

D)

A)

B)

C)

D)

Example 3

Practice 1

y-intercept? __________________

slope?______________________

Find slope from graph and equation. Choose two EXACT points to determine:

Equation: ________________________________________________

38

Notice, each increment in the graphs to the left is…

Practice 2

y-intercept? _________________ slope?______________________

Find slope from graph and equation. Choose two EXACT points to determine:

Equation: _________________________________________________

YOU TRY!

Write the equation of each line. Find slope using the graph and the formula. Then write your answer in slope-intercept form

1. 2.

ANSWER_________________________ ANSWER_________________________

3. 4.

ANSWER_________________________ ANSWER_________________________

39

5. 6.

ANSWER_________________________ ANSWER_________________________

Scatter Plots

40

1)

2)

3)

41

Objective 131a: _________________________________________________________________________________

THREE COMMON FORMS OF LINEAR EQUATIONS

REARRANGING GOALS: 1. y is on ____________side of the equation.

2. The constant (______________) and x (_____________) are on right side.

For each equation below, write what form the current equation is in, if any, then convert to slope-intercept form.

Example 1. y – 3x = 5

42

Example 2. 8 x+2 y=−4 Example 3. 2 x−4 y=6

Example 4. In the equation following equation, what does y equal in terms of x?

y – 2 = ½(x + 5)

43

Example 5

Solve for w∈terms of p5 p+2 p=2w−3w+8

Practice converting for slope-intercept form

1) x− y=10 3) 5 x−4 y=10 4) 3 x+6 y=2

REARRANGING PRACTICE1) Solve for r :3 r−4 r=9 t 2) Put∈slope−intercept form : 4( y−3)=2 x

3¿Solve for r :6+3 r−4 r=9 t 4) Put∈slope−intercept form : 4 y−3+8 x=2 x

44

5¿What does mequal∈terms of b ,3b=2 m+5

6) Solve for w∈terms of p 3 p+ p=3 w−4 w+6

YOU TRY!

1¿Solve for r :6+2r−4 r=6 t 2) Put∈slope−∫ form :3 y−9+2 x=3 x

3¿Solve for m∈terms of b ,9 b=3m+4

4) What doesw equal∈terms of p2 p+2 p=3w−4 w+8

45

Critical Thinking

A) B)

C) −6 x−2 y=10

D) y=−2x+2

1) Which of the four functions above has the greatest rate of change? Which has the smallest rate of change?

2) Looking at the table, what is the value where G(x ) is equal to x?

3) Pick one of the four functions above. Create a real-life problem to represent the function. Be sure to describe the rate of change in the problem as it relates to real-life.

46

X -2 -1 2 3G(x) -14 -9 6 11

Objective 131b: _________________________________________________________________________________

GRAPHING THE EQUATION OF A LINE

EXAMPLE 1: EXAMPLE 2:

47

EXAMPLE 3: EXAMPLE 4:

EXAMPLE 5: y+4=2 x EXAMPLE 6: x−2 y=6

EXAMPLE 7: x=−4 EXAMPLE 8: y=2 x

48

YOU TRY!

Graph the following lines

49

1) 2)

4)

5)

6)

7) 8)

9) 10)

11) 12)

50

Contextual Problems

1) David wants to buy a new pair of Jordan’s that cost $150. He is saving up $25 each week. Create a graph to show his savings each week in relation to being able to purchase the new Jordan’s.

2) Write your own word problem, then create a graph.

51

Some Extra Practice

1) Determine whether or not the following points are on the line described by: −2 x+4 y=12. Show your work and explain your rationale for each point.

Point A: (7, -1)

Point B: (-3, 1.5)

2) For problem 1 above, list three other points that are on the line. Prove they are on the line as well.

3) Draw a quick sketch of a coordinate plane below. Draw a line that passes through (-2, 3) and has a slope of -3. Then, draw another coordinate plane and title it “common mistake.” Using the same information above, make what you think would be the most common mistake a student would make. Explain why you think that is the most common mistake.

52

4) Explain the smartcut you know for writing the slope intercept equation from the ordered pairs (0, -4) and (2,10). Be sure to show your work and write the equation as part of your explanation.

5) Benny started at point (-2,3). He walked 20 blocks north and 5 blocks east (assume each block is one unit). He then walked 10 blocks south and 8 blocks west. At what point does Benny now stand? Show all calculations.

6) What is the equation of a line that passes through (4,8) and has an X-intercept of -6?

7) What is the slope of the lines that go through the following points?a. (x,y) and (0,0)

b. (x, -y) and (0,0)

8) A line that always has an x-value of d has what for a slope? What is the equation of the line?

53

Objective 131c: _________________________________________________________________________________(Lesson 4-5 in book)

Example One

When the remains of ancient people are discovered, usually only a few bones are found. Anthropologists can determine a person’s height by using a formula that relates the length of the tibia T (shin bone) to the person’s height H, both measured in centimeters. The formula for males is H=81.7+2.4T and for females is H=72.6+2.5T . Complete the tables below, then graph each set of ordered pairs.

Male

Length of Tibia (cm)

Height(cm)

(T , H )

30.5

34.836.3

37.9

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Female

Length of Tibia (cm)

Height(cm)

(T ,H )

30.5

34.836.3

37.9

Problem 2: Carlos swims every day. He burns an average of 10.6 Calories per minute when swimming.

A) What would be a reasonable y-intercept for this problem? Explain your reasoning.

B) Make a table to show the calories burned for every ten minutes of swimming. Your table should include five data points. Then use your table to graph the relationship.

C) Suppose Carlos wants do burn 350 Calories. Approximately how long must he swim for?

D) Carlos has a goal to burn 1,690 Calories in a week. He practices on Mondays, Tuesdays, Thursdays, and Fridays. How long must he swim each day, on average, to burn those calories in one week? Show all organized work.

55

Problem 3: Mr. Erwin’s grandmother had to call a plumber to fix her rusted out pipes under the 30 year old sink. The plumber charges a $50 house call fee for coming to her house, and then $65 per hour for labor. The cost of materials is added to the bill at the end.

a) What is the y-intercept of this function? Explain.

b) Create a graph to represent the cost of the plumber's labor only fixing her sink, assuming it will take less than five hours.

c) The plumber worked 1.75 hours on her sink. Materials for the sink were $35.10. There is also a 9% tax back in Iowa where Mr. Erwin’s grandmother lives. How much did she pay to have her sink repaired? Show all organized work.

56

Objective 132: ___________________________________________________________________________

Define:

x-intercept:

y-intercept:

Example One: x - 2y = 6

Find the x intercept. Find the y intercept.

x-intercept = ( , ) y-intercept = ( , )

Example Two: -3x + 5y = 9

Find the x intercept. Find the y intercept.

x-intercept = ( , ) y-intercept = ( , )

57

Example Three: y = 7

Find the x intercept. Find the y intercept.

x-intercept = ( , ) y-intercept = ( , )

Example 4: Give an example of an equation with no y intercept.

You try the following. Find the x and y intercepts.

5. 7x + 3y= 21 6. 8x = 16 + 5y 7. 23

x−5 y=15

x-int.: _____ x-int. _____ x-int. _______

y-int: _____ y-int _____ y-int _______

58

8. y=2x - 4 9. y= -3x 10. y= -1/2x + 5

x-int.: _____ x-int. _____ x-int. _______

y-int: _____ y-int _____ y-int _______

11) At a football game, student tickets are $3 and adult tickets are $5. Let x be the number of student tickets sold. Let y be the number of adult tickets sold. The equation 3x + 5y = 630 shows that $630 was raised from ticket sales. Find the intercepts and explain what each means. Then graph the function.

59

12) The trophy case that Mr. Erwin built is on the 2nd floor at Pritzker. The two side sections are three feet wide by 7 feet tall. He is trying to estimate how many small and large trophies will fit in the two side sections. He estimates that small trophies will require 288 square inches of space, and the larger trophies will need 540 square inches of space.

A) Write an equation in standard form to represent how many trophies the two sections combined will hold.

B) He knows he has six large trophies. How many smaller trophies will he be able to put in the two side sections? Show each step, explain why you took each step, and justify your answer.

C) What are the x- and y-intercepts in this problem? Explain what each means in the context of this problem.

D) The middle section of the trophy case is four feet wide by seven feet tall. What would be the new equation in standard form to represent the area of the middle section? As the area of the trophy case increased, what is the effect on the slope and y-intercept of the equation. Explain.

60

13) Write your own word problem, find the intercepts, and interpret what they mean.

61