Slide 1 / 96

Linear Relations and

Functions

Slide 2 / 96

Table of ContentsScatter Plots

Step, Absolute Value, Piecewise, Identity, and Constant Functions

Graphing Inequalities

Slide 3 / 96

Scatter Plots

Return toTable of Contents

Slide 4 / 96

A scatter plot is a graph that shows a set of data that has two variables.

Time Studying

Tes t Score

45 89

30 78

50 90

60 92

40 85

48 87

55 95

35 82What do you observe?

Time spent studying

Test

Sco

re

Slide 5 / 96

Time spent studying

Test

Sco

re

Predict the test score of someone who spends 52 minutes studying

Predict the test score of someone who spends 75 minutes studying

Slide 6 / 96

Predict the height of a person who wears a size 8 shoe

Predict the shoe size of a person who is 50 inches tall

the

shoe size

heig

ht in

inch

es

Shoe size & Height

Slide 7 / 96Notice that the points form a linear like pattern. To draw a line of best fit, use two points so that the line is as close as possible to the data points.

Our line is drawn so that it fits as close as possible to the data points. This line was drawn through (35,82) and (50,90).

Slide 8 / 96

1 Consider the scatter graph to answer the following: Which 2 points would give the best line of fit?

A

B

C

D

X Y

3 9

4.5 8

5 7

6 5

8 4

9 3

10 1

B

CD

A and D

B and C

C and D

there is nopattern

A

Slide 9 / 96

2 Consider the scatter graph to answer the following: Which 2 points would give the best line of fit?

A

B

C

D

X Y

5 2

6 4

7 3

8 4

9 4.5

9 5

10 3

A and D

B and C

C and D

there is nopattern

A

CBD

Slide 10 / 96

Determining the

Prediction Equation

Return toTable ofContents

Slide 11 / 96

The points form a linear like pattern, so use two of the points to draw a line of best fit.

Our line is drawn so that it fits as close as possible to the data points. This line was drawn through (35,82) and (50,90).

Slide 12 / 96

Slide 13 / 96 Slide 14 / 96

Slide 15 / 96 Slide 16 / 96

If a student got an 80 on the test, What would be the predicted length of their study time?

The student studied about 31 minutes.

Slide 17 / 96

3 Consider the scatter graph to answer the following: What is the slope of the line of best fit going through A and D?

A

BC

D

X Y

3 9

5 7

6 5

8 4

9 3

10 1

A

D(9, 3)

(3, 9)

Slide 18 / 96

4 Consider the scatter graph to answer the following: What is the y-intercept of the line of best fit going through A and D?

A

B

C

D

X Y

3 9

4.5 8

5 7

6 5

8 4

9 3

10 1

A

D

(9, 3)

(3, 9)

Slide 19 / 965 Consider the scatter graph to answer the following: The

equation for our line is y = -1x + 12. What would the prediction be if x = 7? Is this an interpolation or extrapolation?

A

B

C

D

X Y

3 9

4.5 8

5 7

6 5

8 4

9 3

10 1

A

D

5, interpolation

5, extrapolation

6, interpolation

6, extrapolation

Slide 20 / 966 Consider the scatter graph to answer the following: The

equation for our line is y = -1x + 12. What would the prediction be if x = 14? Is this an interpolation or extrapolation?

A

B

C

D

X Y

3 9

4.5 8

5 7

6 5

8 4

9 3

10 1

A

D

-4, interpolation

-4, extrapolation

-2, interpolation

-2, extrapolation

Slide 21 / 967 Consider the scatter graph to answer the following: The

equation for our line is y = -1x + 12. What would the prediction be if y = 11? Is this an interpolation or extrapolation?

A

B

C

D

X Y

3 9

4.5 8

5 7

6 5

8 4

9 3

10 1

A

D

1, interpolation

1, extrapolation

2, interpolation

2, extrapolation

Slide 22 / 968 In the previous questions, we began by using the table at

the right. Which of the predicted values: (7,5) or (14, -2) will be more accurate and why?

A

B

C

D

X Y

3 9

4.5 8

5 7

6 5

8 4

9 3

10 1

(7,5); it is an interpolation.

(7,5); there already is a 5 and a 7 in the table

(14, -2) it is an extrapolation

(14, -2); the line is going down and will becomenegative

Slide 23 / 96

Step, Absolute Value, Piecewise, Identity, and

Constant Functions

Return toTable of Contents

Slide 24 / 96

There are special functions that have there own names and graphs.

Constant Function Identity Function

y = bDomain: Reals

Range: {2}

y = xDomain: RealsRange: Reals

Slide 25 / 96

Absolute Value Function

y = a|cx -h| + kDomain: RealsRange: -2 < y

Slide 26 / 96

To Graph an Absolute Value Graph1) Set the value inside of the absolute value sign equal to zero and solve.This is x-value of the vertex of the graph.

2) Create a table. Use the solution to step one as a middle value by picking a couple of points smaller and a couple larger. Complete the table.

3) Graph the points.

4) Connect the points.

5) As a check, if the number in front of the absolute value sign is positive the "V" opens up, if its negative it opens down.

X Y

1 0

2 -2

3 -4

4 -2

5 0

D:Reals; R: -4 < y

Slide 27 / 96

Graph y = -3| 2x + 4| + 5

X Y

D: _______ ; R:__________

Slide 28 / 96

Graph y = 2| -2x + 6| - 2

X Y

D: _______ ; R:__________

Slide 29 / 96

9 What is the vertex of y = | 2x -1| +2

A -2

B 1

C 1/2

D 2

Slide 30 / 96

10 Which of the following is the correct graph of y = | x+4| - 2 ?

A B

C D

Slide 31 / 96

11 Which of the following is the correct graph of y = | x - 4| - 2 ?

A B

C D

Slide 32 / 96

12 Which of the following is the correct graph of y = -2 | 3x + 9| + 5 ?

A B

C D

Slide 33 / 96

13 Graph y = |2x - 6| + 3

Slide 34 / 96

14 What is the domain of the graphed function?

A Set of Integers

B Set of Reals

C x > -3

D x < -3

Slide 35 / 96

15 What is the range of the graphed function?

A Set of Integers

B Set of Reals

C y > -3

D y < -3

Slide 36 / 96

16 What is the domain of the graphed function?

A Set of Integers

B Set of Reals

C x > 3

D x < 3

Slide 37 / 96

17 What is the domain of the graphed function?

A Set of Integers

B Set of Reals

C y > 3

D y < 3

Slide 38 / 96

Greatest Integer Functions

[2] = 2[2.1] = 2[2.3] = 2[2.5] = 2[2.75] = 2[2.999] = 2[3] = 3

[-2] = -2[-2.1] = -3[-2.3] = -3[-2.5] = -3[-2.75] = -3[-2.999] = -3[-3] = -3

The [ ] tell you to round to the preceding integer. Think round to the left on a number line.

[ ] are a grouping sign and the inside should be simplified before rounding.

Slide 39 / 96

Evaluate

[3.5 + .6] 4

[3.7 - .8] 2

[ 2 - 2.1] -1

3[2.4 +.2] 6

[3(2.4) + .2] 7

3[2.4] + .2 6.2

4[2.1 - 2]2 0

Slide 40 / 96

18 Evaluate [2.6]

Slide 41 / 96

19 Evaluate [5+2]

Slide 42 / 96

20 Evaluate [ -2.6 ]

Slide 43 / 96

21 Evaluate [ -2.1 ]

Slide 44 / 96

22 Evaluate 3[2.6 + .5]2

Slide 45 / 96

Graphing a Greatest Integer Function

It also called a StepFunction because ofthe shape of its graph.

Domain: RealsRange: Integers

Slide 46 / 96

Graphing a Greatest Integer Function

1) Find the values of x that don't have to be rounded. The inside of [ ] determines that.

2) Make a table. Pick values around the integer values in step 1.Remember our graph will look like steps so once we know the height and width of each step we can repeat the pattern.

3) Graph. Continue the pattern to complete.

X Y

0 0

0.2 0

0.4 0

0.5 1

0.8 1

0.9 1

1 2

1.5 3(graph is on next page)

Slide 47 / 96

X Y

0 0

0.2 0

0.4 0

0.5 1

0.8 1

0.9 1

1 2

1.5 3

Slide 48 / 96

Graph y = [x +1]

X Y

Slide 49 / 96

Graph y = [4x]

X Y

Slide 50 / 96

Graph y = 2[x -3]

X Y

Slide 51 / 96

Graph y = [.5x]

X Y

Slide 52 / 96

Graph y = 2[.5x + 1]

X Y

Slide 53 / 96

23 What is the domain of the graphed function?

A Set of Integers

B Set of Reals

C Set of Odd Integers

D Set of Even Integers

Slide 54 / 96

24 What is the range of the graphed function?

A Set of Integers

B Set of Reals

C Set of Odd Integers

D Set of Even Integers

Slide 55 / 96

25 What is the domain of the graphed function?

A Set of Integers

B Set of Reals

C Set of Odd Integers

D Set of Even Integers

Slide 56 / 96

26 What is the range of the graphed function?

A Set of Integers

B Set of Reals

C Set of Odd Integers

D Set of Even Integers

Slide 57 / 96

Piecewise Function

A cell phone carrier charge $70 for the first 1000 minutes and $.25 for each minute after the first 1000.

The graph starts as a constant graph of y = 70 and after x = 1000 the graph becomes y = .25(x - 1000) +70

In words:

In Mathematical Notation:

Slide 58 / 96

Piecewise Function

A piecewise function is a combination of other functions.

A cell phone carrier charge $70 for the first 1000 minutes and $.25 for each minute after the first 1000.

The graph starts as a constant graph of y = 70 and after x = 1000 the graph becomes y = .25(x - 1000) +70

Slide 59 / 96

Create the piecewise notation for the following graph.

if

if

Slide 60 / 96

What is the domain and range of the function?

Domain: RealsRange: y < 1

Domain:

Range:

Slide 61 / 96

Find the following values:

x= -2 y=

x= 0 y=

x= 4 y=

Slide 62 / 96

Create the piecewise notation for the following graph.

if

if

Slide 63 / 96

What are the domain and range of the function?

Domain: RealsRange: Reals

Domain:

Range:

Slide 64 / 96

Find the following

f(-2)=

f(0)=

f(2)=

Slide 65 / 96

Create the piecewise notation for the following graph.

if

if

if

Slide 66 / 96

State the domain and range of the function.

Domain: x < 2 or x > 3Range: {-3, -1, 1}

Domain:

Range:

Slide 67 / 96

Find the following values

f(-3)=

f(0)=

f(2.5)=

f(4)=

Slide 68 / 96Graphing a piecewise function:

· Visualize the entire graphof each individual y=

· Graph only the parts defined by x(Be aware of endpoints<open or closed>)

· Repeat for each part.

Slide 69 / 96Graphing a piecewise function:

· Visualize the entire graphof each individual y=

· Graph only the parts defined by x(Be aware of endpoints<open or closed>)

· Repeat for each part.

Slide 70 / 96Graphing a piecewise function:

· Visualize the entire graphof each individual y=

· Graph only the parts defined by x(Be aware of endpoints<open or closed>)

· Repeat for each part.

Slide 71 / 96

What is the domain and range of the function?

Domain: RealsRange: y = -4 or 1 < y < 5

Slide 72 / 96

Graph:

Slide 73 / 96

Graph:

Slide 74 / 96

Slide 75 / 96

Graph:

Slide 76 / 96

Graphing Inequalities

Return toTable of Contents

Slide 77 / 96

A linear inequality, such as y<2x-1,will be represented by a region of points,not just a line.

It does have some similarities to our linear equations, like y=2x-1, it also goes beyond the line.

Slide 78 / 96

The following are linear inequalities.y>mx+b y<mx+b

y<mx+by>mx+b

How do the graphs at leftcompare with y=mx+b?

Next slide for observations.

Slide 79 / 96

The following are linear inequalities.y>mx+b y<mx+b

y<mx+by>mx+b

How do the graphs at leftcompare with y=mx+b?

Shading is above a dotted line.This means the answers are above the line but not on it.

Shading is above a solid line.This means the answers are above the line and on it.

Shading is below a solid line.This means the answers are below the line and on it.

Shading is below a dotted line.This means the answers are below the line but not on it.

Slide 80 / 96

How to Graph a Linear Inequality

1) Decide where the boundary goes.Solve inequality for y, for example y>2x-1.

2)Decide whether boundary should be solid(< or >: points on the boundary make the inequality true) or dashed(< or >: points on the boundary make the inequality false).

3) Graph the bounds.

4) Decide where to shade: y> or y>: shade above the boundary (refer to y-axis) y< or y<: shade below the boundary (refer to y-axis) (or you can test a point, which will be explained later.)

Thinky=mx+b

to graph the boundary

Slide 81 / 96

Graph y<-2x+1

Solution: Think y=-2x+1, m=-2 and b=1.The line should be dotted, so we can graph the boundary line.

Now decide on shading. Since y< we shade below.

Slide 82 / 96

Graph 2x-y<4

Solution: First solve for y. 2x-y < 4 -y < -2x+4 y > 2x-4 (divided by -1)So boundary has m=2 and b=-4, and since the inequality is y> we use a solid line.

Now decide on shading. Since it is y> we shade above.

Slide 83 / 96

Graph y > 1/2x -2.

Solution: Equation is already solved for y som=1/2 and b= -2. The boundary should be a dashed line since the inequality is y>

Now decided on shading. Since y> is beinggraphed, shade above.

Slide 84 / 96

Graph y-2 < -2(x+1)

Solution: First solve for y.y-2 < -2(x+1)y-2 < -2x -2y< -2x -2+2y<-2x so m=-2 and b=0 and since its y< a dashed line is needed.

To finish graphing we shade below the boundary.

Slide 85 / 96

27 For which of these equations would the graph have a solid bounds and be shaded above?

A y < 3x-2B y < 3x-2C y > 3x-2

D y > 3x-2

Slide 86 / 96

28 For which of these equations would the graph have a dashed bounds and be shaded above?

A y < 3x-2B y < 3x-2C y > 3x-2

D y > 3x-2

Slide 87 / 96

29 Which inequality is graphed?

A y < 3x-2B y < 3x-2C y > 3x-2

D y > 3x-2

Slide 88 / 96

30 Which inequality is graphed?

A y < 3x-2B y < 3x-2C y > 3x-2

D y > 3x-2

Slide 89 / 96

31 Why are some bounds dashed?

A Becasue we need to know where to stop shading.

B Becasue points on the line make the inequality false.C it depends on the slope of the line.D I don't know why.

Slide 90 / 96

Inequalities can be graphed without converting to slope y-intercept form.

The steps to graph look the same, the one thing thatchanges is determining where to shade.

1)Decide if bounds is solid or dashed.2)Graph bounds3) Decide where to shade: test a point

Testing a point: Since the shaded region represents all of the points that make the inequality true, if your test point makes the inequality true shade that region.

Slide 91 / 96Graph y-2>3(x-1)

Solution: The bounds needs to be dashed.

Test a point. A point needs to be test that is not on the bounds since we're deciding which region to shade. To make this as easy as possible pick a point with as many zeroes as you can, the origin is great if the line doesn't go through it.Test (0,0): 0-2>3(0-1) -2>-3 this is a true statement so shade the region with (0,0).

Slide 92 / 96

Graph y+3> 2(x+4)

Solution: Bounds is solid.

Test a point: (0,0): 0+3 > 2(0+4) 3 > 8 this is false.Since (0,0) made a false statement shade the region that does not contain the origin.

Slide 93 / 96

Graph y > -4x

Solution: The bounds is dashed.

Test a point: Since the line goes through the origin we need to test a different point. Test a point: (2,0) 0>-4(2)0>-8 this is true, so shade the region with (2,0)

Slide 94 / 96

32 Given the equation y-2 > 1/2(x+4), which test point should be used and where should the graph be shaded?A (0,0): shade above boundary

B (0,0); shade below boundaryC (0,4): shade above boundaryD (0,4); shade below boundary

(-4,2)

(0,0)

Slide 95 / 96

33 Given the equation y < 1/2x, which test point should be used and where should the graph be shaded?

A (0,0): shade above boundary

B (0,0); shade below boundaryC (0,4): shade above boundaryD (0,4); shade below boundary

(0,0)

(0,4)

Slide 96 / 96

34 What point(s) can be used as test points?

A Only the origin.

B Any point with a zero in the ordered pair.C Any point.D Any point not on the boundary.