4.1 plot points in a coordinate plane - rjs solutions · lah_a1_11_fl_ntg_ch04_85-118.indd 89...

VOCABULARY

Quadrant

Goal p Identify and plot points in a coordinate plane.

Copyright © Holt McDougal. All rights reserved. Lesson 4.1 • Algebra 1 Notetaking Guide 85

4.1 Plot Points in a Coordinate Plane

Give the coordinates of the point.

a. A b. B

Solution

a. Point A is units to the

x

y

1

3

12121

23

23 3

AB

CD

E

of the origin and units .

The x-coordinate is .The y-coordinate is .The coordinates are .

b. Point B is units to the of the origin and

units .The x-coordinate is .The y-coordinate is .The coordinates are .

Example 1 Name points in a coordinate plane

Points in Quadrant l have two positive coordinates. Points in the other three quadrants have at least one negative coordinate.

1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.

Checkpoint Complete the following exercise.

Your Notes

LAH_A1_11_FL_NTG_Ch04_85-118.indd 85LAH_A1_11_FL_NTG_Ch04_85-118.indd 85 3/2/09 4:21:55 PM3/2/09 4:21:55 PM

VOCABULARY

Quadrant One of the four parts into which the axes divide a coordinate plane; labeled I, II, III, IV

Goal p Identify and plot points in a coordinate plane.


4.1 Plot Points in a Coordinate Plane

Give the coordinates of the point.

a. A b. B

Solution

a. Point A is 2 units to the

x

y

1

3

12121

23

23 3

AB

CD

E

right of the origin and 2 units up .The x-coordinate is 2 .The y-coordinate is 2 .The coordinates are (2, 2) .

b. Point B is 4 units to the left of the origin and 2 units up .The x-coordinate is 24 .The y-coordinate is 2 .The coordinates are (24, 2) .

Example 1 Name points in a coordinate plane

Points in Quadrant l have two positive coordinates. Points in the other three quadrants have at least one negative coordinate.

1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.

C(22, 23), D(4, 23), E(1, 24)


Your Notes


Plot the point in a coordinate plane. Describe the location of the point.

a. A(0, 3) b. B(1, 22) c. C(23, 24)

Solution

a. Begin at the .

x

y

1

3

12121

23

23 3

Move units .Point A is on the .

b. Begin at the . Move unit to the .Move units .Point B is in Quadrant .

c. Begin at the Move units to the .Move units .Point C is in Quadrant .

Example 2 Plot points in a coordinate plane

Checkpoint Plot the point in a coordinate plane. Describe the location of the point.

2. A(24, 24) 3. B(2, 0)

x

y

1

3

12121

23

23 3

x

y

1

3

12121

23

23 3

86 Lesson 4.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes


Plot the point in a coordinate plane. Describe the location of the point.

a. A(0, 3) b. B(1, 22) c. C(23, 24)

Solution

a. Begin at the origin .

x

y

1

3

12121

23

23 3

A(0, 3)

C(23, 24)

B(1, 22)

Move 3 units up .Point A is on the y-axis .

b. Begin at the origin . Move 1 unit to the right .Move 2 units down .Point B is in Quadrant lV .

c. Begin at the origin. Move 3 units to the left .Move 4 units down .Point C is in Quadrant lIl .

Example 2 Plot points in a coordinate plane

Checkpoint Plot the point in a coordinate plane. Describe the location of the point.

2. A(24, 24) 3. B(2, 0)

x

y

1

3

12121

23

23 3

A(24, 24)

x

y

1

3

12121

23

23 3

B(2, 0)

Quadrant III x-axis


Your Notes


Graph the function y 5 x 1 1 with domain 22, 21, 0, 1, 2. Then identify the range of the function.

Solution x y 5 x 1 1

22 y 5 22 1 1 5

21 y 5 21 1 1 5

0 y 5 0 1 1 5

1 y 5 1 1 1 5

2 y 5 2 1 1 5

Step 1 Make a table.

Step 2 List the ordered pairs:

(22, ), (21, ), (0, ), (1, ), (2, ).

Then graph the function.

x

y

1

3

12121

23

23 3

Step 3 Identify the range: .

Example 3 Graph a function

Homework


4. Graph the function y 5 2 1 } 2 x 1 3 with domain

24, 22, 0, 2, and 4. Then identify the range.

x

y

1

3

5

12121

23

23 3


Your Notes


Graph the function y 5 x 1 1 with domain 22, 21, 0, 1, 2. Then identify the range of the function.

Solution x y 5 x 1 1

22 y 5 22 1 1 5 21

21 y 5 21 1 1 5 0

0 y 5 0 1 1 5 1

1 y 5 1 1 1 5 2

2 y 5 2 1 1 5 3


Step 2 List the ordered pairs:

(22, 21 ), (21, 0 ), (0, 1 ), (1, 2 ), (2, 3 ).

Then graph the function.

x

y

1

3

12121

23

23 3

Step 3 Identify the range: 21, 0, 1, 2, 3 .


Homework


4. Graph the function y 5 2 1 } 2 x 1 3 with domain

24, 22, 0, 2, and 4. Then identify the range.

x

y

1

3

5

12121

23

23 3

range: 1, 2, 3, 4, 5


Your Notes


Perform Transformations Goal p Perform and describe transformations in a

coordinate plane.

VOCABULARY

Transformation

Translation

Vertical stretch

Vertical shrink

Reflection

The graphs show the function y 5 x 2 1 with domain 1, 2, 3. (a) The transformation (x, y) → (x, y 1 2) moves the graph units. (b) The transformation (x, y) → (x, 23y) is a vertical a reflection in the x-axis Draw the transformed graphs.

(a). y

x1

1

(b). y

x

1

1

Original Image Original Image

( , 0) → ( ) ( , ) → ( , )

→ →

→ →

Example 1 Perform transformations

The functions represented by the original and both image graphs have the same domain. The range for the original function is 1, 2, . The ranges for image graph (a) and (b) are respectively, 2, , and , , .

88 4.1 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Focus On GraphingUse after Lesson 4.1

Your Notes


Perform Transformations Goal p Perform and describe transformations in a

coordinate plane.

VOCABULARY

Transformation Applying a rule to the coordinates of each point on a graph to produce an image

Translation Moves every point on a graph the same distance in the same direction

Vertical stretch Moves a point away from the x-axis

Vertical shrink Moves a point toward the x-axis

Reflection Flips a point in a line

The graphs show the function y 5 x 2 1 with domain 1, 2, 3. (a) The transformation (x, y) → (x, y 1 2) moves the graph up 2 units. (b) The transformation (x, y) → (x, 23y) is a vertical stretch with a reflection in the x-axis Draw the transformed graphs.

(a). y

x1

1

(b). y

x

1

1

Original Image Original Image

( 1 , 0) → ( 1 , 2 ) ( 1 , 0 ) → ( 1 , 0 )

(2, 1) → (2, 3) (2, 1) → (2, 23)

(3, 2) → (3, 4) (3, 2) → (3, 26)

Example 1 Perform transformations

The functions represented by the original and both image graphs have the same domain. The range for the original function is 1, 2, 3 . The ranges for image graph (a) and (b) are respectively, 2, 3 , 4 and 0 , 23, 26.



Your Notes


1. Use small dots to graph the function: y 5 3x 2 1 with domain 21, 0, 1. Perform the transformation (x, y) → (x, y 2 2). Graph the image using large dots. Identify the domain and range of the function represented by the image.

2. Use small dots to graph the

function: y 5 22x 1 2 with domain 22, 0, 2. Perform the transformation (x, y) → (x, 0.5y). Graph the image using large dots. Identify the domain and range of the function represented by the image.

Copyright © Holt McDougal. All rights reserved. 4.1 Focus on Graphing • Algebra 1 Notetaking Guide 89

y

x1

1

IDENTIFYING TRANSFORMATIONS

Vertical or

(x, y) → (x1h, y ) (x, y) → (x, y 1 k)

reflection(x, y) → (x, ay)

where a 0

reflection

(x, y) → (x, ay)

where a 0y

x1

1

y

x1

1

y

x1

1

y

x1

1

Checkpoint Complete the following exercises.

y

x

1

1

Homework

Your Notes


1. Use small dots to graph the function: y 5 3x 2 1 with domain 21, 0, 1. Perform the transformation (x, y) → (x, y 2 2). Graph the image using large dots. Identify the domain and range of the function represented by the image.

domain: 21, 0, 1;

range: 26, 23, 0

2. Use small dots to graph the

function: y 5 22x 1 2 with domain 22, 0, 2. Perform the transformation (x, y) → (x, 0.5y). Graph the image using large dots. Identify the domain and range of the function represented by the image.

domain: 22, 0, 2 range: 3, 1, 21


y

x1

1

IDENTIFYING TRANSFORMATIONS

Translation Vertical stretch or shrink

Horizontal (x, y) → (x1h, y )

Vertical (x, y) → (x, y 1 k)

Without reflection

(x, y) → (x, ay)

where a . 0

With reflection

(x, y) → (x, ay)

where a , 0y

x1

1

y

x1

1

y

x1

1

y

x1

1


y

x

1

1

Homework

Your Notes


4.2 Graph Linear EquationsGoal p Graph linear equations in a coordinate plane.

VOCABULARY

Solution of an equation in two variables

Graph of an equation in two variables

Linear equation

Standard form of a linear equation

Linear function

Your Notes

Graph the equation x 1 y 5 4.

SolutionStep 1 Solve the equation for y.

x 1 y 5 4

y 5


Choose a few values for x and find the values for y.

x 22 21 0 1 2

y

Example 1 Graph an equation

Use convenient values for x when making a table. These should include a combination of negative values, zero, and positive values.



4.2 Graph Linear EquationsGoal p Graph linear equations in a coordinate plane.

VOCABULARY

Solution of an equation in two variables An ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the equation

Graph of an equation in two variables The set of points in a coordinate plane that represents all solutions of the equation

Linear equation An equation whose graph is a line

Standard form of a linear equation Ax 1 By 5 C, where A, B, and C are real numbers and A and B are not both zero

Linear function The equation Ax 1 By 5 C represents a linear function provided B Þ 0 (that is, provided the graph of the equation is not a vertical line).

Your Notes

Graph the equation x 1 y 5 4.

SolutionStep 1 Solve the equation for y.

x 1 y 5 4

y 5 4 2 x


Choose a few values for x and find the values for y.

x 22 21 0 1 2

y 6 5 4 3 2

Example 1 Graph an equation

Use convenient values for x when making a table. These should include a combination of negative values, zero, and positive values.



Step 3 Plot the points.

x

y

1

3

5

12121

23 3

Step 4 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

Your Notes


Graph (a) y 5 23 and (b) x 5 2.

Solutiona. Regardless of the value of x, the value of y is always

. The graph of y 5 23 is a line 3 units the x-axis.

x

y

1

3

12121

23

23 3

b. Regardless of the value of y, the value of x is always . The graph of x 5 2 is a line

2 units to the of the y-axis.

x

y

1

3

12121

23

23 3

Example 2 Graph y 5 b and x 5 a

The equations y 5 23 and 0x 1 1y 5 23 are equivalent. For any value of x, the ordered pair (x, 23) is a solution of y 5 23.


Step 3 Plot the points.

x

y

1

3

5

12121

23 3

Step 4 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

Your Notes


Graph (a) y 5 23 and (b) x 5 2.

Solutiona. Regardless of the value of x, the value of y is always

23 . The graph of y 5 23 is a horizontal line 3 units below the x-axis.

x

y

1

3

12121

23

23 3

y 5 23

b. Regardless of the value of y, the value of x is always 2 . The graph of x 5 2 is a vertical line 2 units to the right of the y-axis.

x

y

1

3

12121

23

23 3

x 5 2

Example 2 Graph y 5 b and x 5 a

The equations y 5 23 and 0x 1 1y 5 23 are equivalent. For any value of x, the ordered pair (x, 23) is a solution of y 5 23.


1. y 5 2x 2 1 2. x 5 0.5

x

y

1

3

12121

23

25

23 3

x

y

1

3

12121

23

23 3

3. y 5 24x 1 1 4. y 5 21.5

x

y

2

6

12122

26

23 3

x

y

1

3

12121

23

23 3

Checkpoint Graph the equation.

EQUATIONS OF HORIZONTAL AND VERTICAL LINES

1. The graph of y 5 b is a line.

2. The line of graph y 5 b passes through the point .

3. The graph of x 5 a is a line.

4. The line of graph x 5 a passes through the point .


Your Notes


1. y 5 2x 2 1 2. x 5 0.5

x

y

1

3

12121

23

25

23 3

x

y

1

3

12121

23

23 3

x 5 0.5

3. y 5 24x 1 1 4. y 5 21.5

x

y

2

6

12122

26

23 3

x

y

1

3

12121

23

23 3

Checkpoint Graph the equation.

EQUATIONS OF HORIZONTAL AND VERTICAL LINES

1. The graph of y 5 b is a horizontal line.

2. The line of graph y 5 b passes through the point (0, b) .

3. The graph of x 5 a is a vertical line.

4. The line of graph x 5 a passes through the point (a, 0) .


Your Notes


Graph the function y 5 2x 1 2 with domain x ≥ 0. Then identify the range of the function.

Solution

Step 1 Make a .

x 0 1 2 3 4

y

Step 2 Plot the .

x

y

1

3

5

7

9

121 3 5

Step 3 Connect the points with a because the domain is .

Step 4 Identify the range. From the graph, you can see that all points have a y-coordinate of , so the range of the function is .

Example 3 Graph a linear function


Homework

5. Graph the function y 5 2x 1 4 with domain x $ 0. Then identify the range of the function.

x

y

1

3

12121

23

23 3


Your Notes


Graph the function y 5 2x 1 2 with domain x ≥ 0. Then identify the range of the function.

Solution

Step 1 Make a table .

x 0 1 2 3 4

y 2 4 6 8 10

Step 2 Plot the points .

x

y

1

3

5

7

9

121 3 5

Step 3 Connect the points with a ray because the domain is restricted .

Step 4 Identify the range. From the graph, you can see that all points have a y-coordinate of 2 or more , so the range of the function is y ≥ 2 .

Example 3 Graph a linear function


Homework

5. Graph the function y 5 2x 1 4 with domain x $ 0. Then identify the range of the function.

x

y

1

3

12121

23

23 3

range: y ≤ 4


Your Notes


Identify Discrete and Continuous Functions Goal p Graph and classify discrete and continuous

functions.

VOCABULARY

Discrete function

Continuous function

DISCRETE AND CONTINUOUS FUNCTIONS

The graph of a The graph of a

consists of isolated points. is unbroken.

y

x

y

x

Graph the function y 5 3x 2 2 with the given domain. Classify the function as discrete or continuous. Then identify the range of the function.a. Domain: x 5 21, 0, 1 b. Domain: x # 1

y

x1

1

y

x1

1

The graph consists of The graph is , points, so so the function is

the function is . The range is . Its range is .

Example 1 Graph and classify a function

94 4.2 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Focus On FunctionsUse after Lesson 4.2

Your Notes


Identify Discrete and Continuous Functions Goal p Graph and classify discrete and continuous

functions.

VOCABULARY

Discrete function A function that has a graph consisting of isolated points

Continuous function A function that has an unbroken graph

DISCRETE AND CONTINUOUS FUNCTIONS

The graph of a The graph of a discrete function continuous function consists of isolated points. is unbroken.

y

x

y

x

Graph the function y 5 3x 2 2 with the given domain. Classify the function as discrete or continuous. Then identify the range of the function.a. Domain: x 5 21, 0, 1 b. Domain: x # 1

y

x1

1

y

x1

1

The graph consists of The graph is unbroken , isolated points, so so the function isthe function is discrete . continuous The range is 25, 22, 1 . Its range is y # 1 .

Example 1 Graph and classify a function

94 4.2 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Focus On FunctionsUse after Lesson 4.2

Your Notes


1. Graph the function y 5 x 2 3 with the domain: x $ 2. Classify the function as discrete or continuous. Then identify the range.


Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, graph the function and find the value of y when x 5 0.5.

Time after planting (days), x 1 2 3Growth of plant (millimeters), y 14 28 42

Solution

It make sense to talk about the amount a plant has grown after any length of time. So the table represents a function.

Example 2 Classify and graph a real-world function

Copyright © Holt McDougal. All rights reserved. 4.2 Focus On Functions• Algebra 1 Notetaking Guide 95

y

x1

1

y

x0 1 2 3 4

10

20

30

40

Plant Growth

Time after planting(days)

Gro

wth

of

pla

nt

(mill

imet

ers)

2. Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, find the value of y when x 5 2.5. Round your answer to the nearest hundredth.

Number of songs downloaded, x 1 5 10

Cost of songs (dollars), y 1.25 6.25 12.50


Homework

Your Notes


1. Graph the function y 5 x 2 3 with the domain: x $ 2. Classify the function as discrete or continuous. Then identify the range.

Continuous; y $ −1


Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, graph the function and find the value of y when x 5 0.5.

Time after planting (days), x 1 2 3Growth of plant (millimeters), y 14 28 42

Solution

It does make sense to talk about the amount a plant has grown after any length of time. So the table represents a continuous function. When x 5 0.5 days, y 5 7 millimeters

Example 2 Classify and graph a real-world function

Copyright © Holt McDougal. All rights reserved. 4.2 Focus On Functions• Algebra 1 Notetaking Guide 95

y

x1

1

y

x0 1 2 3 4

10

20

30

40

Plant Growth

Time after planting(days)

Gro

wth

of

pla

nt

(mill

imet

ers)

2. Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, find the value of y when x 5 2.5. Round your answer to the nearest hundredth.

Number of songs downloaded, x 1 5 10

Cost of songs (dollars), y 1.25 6.25 12.50

Discrete; you cannot buy part of a song.


Homework

Your Notes


4.3 Graph Using InterceptsGoal p Graph a linear equation using intercepts.

VOCABULARY

x-intercept

y-intercept

Your Notes

Find the x-intercept and the y-intercept of the graph of 8x 2 2y 5 32.

Solution

1. Substitute for y and solve for x.

8x 2 2y 5 32 Write originalequation.

8x 2 2( ) 5 32 Substitute for y.

x 5 5 Solve for .

2. Substitute for x and solve for y.


8( ) 2 2y 5 32 Substitute for x.

y 5 5 Solve for .

The x-intercept is . The y-intercept is .

Example 1 Find the intercepts of the graph of an equation



4.3 Graph Using InterceptsGoal p Graph a linear equation using intercepts.

VOCABULARY

x-intercept The x-coordinate of a point where a graph crosses the x-axis

y-intercept The y-coordinate of a point where a graph crosses the y-axis

Your Notes

Find the x-intercept and the y-intercept of the graph of 8x 2 2y 5 32.

Solution

1. Substitute 0 for y and solve for x.


8x 2 2( 0 ) 5 32 Substitute 0 for y.

x 5 8

32 5 4 Solve for x .

2. Substitute 0 for x and solve for y.


8( 0 ) 2 2y 5 32 Substitute 0 for x.

y 5 22

32 5 216 Solve for y .

The x-intercept is 4 . The y-intercept is 216 .

Example 1 Find the intercepts of the graph of an equation



Checkpoint Find the x-intercept and y-intercept of the graph of the equation.

1. 2x 1 3y 5 18 2. 212x 2 4y 5 36

Graph 3.5x 1 2y 5 14. Label the points where the line crosses the axis.

Solution

Step 1 Find the .

3.5x 1 2y 5 14 3.5x 1 2y 5 14

3.5x 1 2( ) 5 14 3.5( ) 1 2y 5 14

x 5 5

y 5 5

Step 2 Plot the points that correspond to the intercepts.

The x-intercept is , so plot the point .

The y-intercept is , so plot the point .

Step 3 the points by drawing a line through them.

x

y

1

3

5

7

12121

23 3 5

CHECK

You can check the graph of the equation by using a third point. When x 5 2, y 5 , so the ordered pair

is a third solution of the equation. You can see that lies on the graph, so the graph is correct.

Example 2 Use intercepts to graph an equation


Your Notes


Checkpoint Find the x-intercept and y-intercept of the graph of the equation.

1. 2x 1 3y 5 18 2. 212x 2 4y 5 36

x-intercept: 9 x-intercept: 23 y-intercept: 6 y-intercept: 29

Graph 3.5x 1 2y 5 14. Label the points where the line crosses the axis.

Solution

Step 1 Find the intercepts .

3.5x 1 2y 5 14 3.5x 1 2y 5 14

3.5x 1 2( 0 ) 5 14 3.5( 0 ) 1 2y 5 14

x 5

3.5

14 5 4

y 5

2

14 5 7

Step 2 Plot the points that correspond to the intercepts.

The x-intercept is 4 , so plot the point (4, 0) .

The y-intercept is 7 , so plot the point (0, 7) .

Step 3 Connect the points by drawing a line through them.

x

y

1

3

5

7

12121

23 3 5

CHECK

You can check the graph of the equation by using a third point. When x 5 2, y 5 3.5 , so the ordered pair (2, 3.5) is a third solution of the equation. You can see that (2, 3.5) lies on the graph, so the graph is correct.

Example 2 Use intercepts to graph an equation


Your Notes


3. Graph 2x 2 7y 5 14. Label the points where the line crosses the axes.

x

y

1

3

12121

23

3 5 7

4. Identify the x-intercept and y-intercept of the graph.

x

y

1

3

212321

3


Homework

Identify the x-intercept and y-intercept of the graph.

x

y

1

3

12121

23

23 3

SolutionTo find the x-intercept, look to see where the graph crosses the . The x-intercept is . To find the y-intercept, look to see where the graph crosses the . The y-intercept is .

Example 3 Use a graph to find the intercepts


Your Notes


3. Graph 2x 2 7y 5 14. Label the points where the line crosses the axes.

x

y

1

3

12121

23

3 5 7

(7, 0)

(0, 22)

4. Identify the x-intercept and y-intercept of the graph.

x

y

1

3

212321

3

x-intercept 5 1

y-intercept 5 3


Homework

Identify the x-intercept and y-intercept of the graph.

x

y

1

3

12121

23

23 3

SolutionTo find the x-intercept, look to see where the graph crosses the x-axis . The x-intercept is 22 . To find the y-intercept, look to see where the graph crosses the y-axis . The y-intercept is 2 .

Example 3 Use a graph to find the intercepts


Your Notes


4.4 Find Slope and Rate of ChangeGoal p Find the slope of a line and interpret slope as a

rate of change.

VOCABULARY

Slope

Rate of change

FINDING THE SLOPE OF A LINE

Words Symbols

The slope of the nonvertical line m 5 y2 2 y1 } x2 2 x1

passing through the two points (x1, y1) and (x2, y2) is the ratio of the (change in y) to the

(change in x).

slope 5 5 change in y

} change in x

Graph

x

y

(x1, y1)

(x2, y2)

runx22 x1

risey22 y1


Your Notes


4.4 Find Slope and Rate of ChangeGoal p Find the slope of a line and interpret slope as a

rate of change.

VOCABULARY

Slope The ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on a nonvertical line; Represented by m

Rate of change Compares a change in one quantity to a change in another quantity

FINDING THE SLOPE OF A LINE

Words Symbols

The slope of the nonvertical line m 5 y2 2 y1 } x2 2 x1

passing through the two points (x1, y1) and (x2, y2) is the ratio of the rise (change in y) to the run (change in x).

slope 5 run

rise 5

change in y } change in x

Graph

x

y

(x1, y1)

(x2, y2)

runx22 x1

risey22 y1


Your Notes


Find the slope of the line shown.

a. Let (x1, y1) 5 (21, 2) b. Let (x1, y1) 5 (1, 4) and (x2, y2) 5 (3, 5). and (x2, y2) 5 (3, 22).

x

y

1

3

5

12121

23 3

x

y

1

3

5

12121

23

23 3 5

Solution

a. m 5 y2 2 y1 } x2 2 x1

Write formula for slope.

5 2 2

2 (21) Substitute.

5 Simplify. The line from left to right. The slope is

.

b. m 5 y2 2 y1 } x2 2 x1


5 2 4

2 1 Substitute.

5 5 Simplify. The line from left to right. The slope is

.

Example 1 Find a slope

Checkpoint Find the slope of the line passing through the points.

1. (23, 21) and (22, 1) 2. (26, 3) and (5, 22)

Keep the x- and y-coordinates in the same order in the numerator and denominator when calculating slope. This will help avoid error.


Your Notes




x

y

1

3

5

12121

23 3

3

4

m 5 34

x

y

1

3

5

12121

23

23 3 5

2

26 m 5 23

Solution

a. m 5 y2 2 y1 } x2 2 x1


5

5 2 2

3 2 (21) Substitute.

5 3 } 4 Simplify. The line rises from left to right. The slope is

positive .

b. m 5 y2 2 y1 } x2 2 x1


5

22 2 4

3 2 1 Substitute.

5 26 } 2 5 23 Simplify. The line falls from left to right. The slope is

negative .

Example 1 Find a slope

Checkpoint Find the slope of the line passing through the points.

1. (23, 21) and (22, 1) 2. (26, 3) and (5, 22)

2 2 5 } 11

Keep the x- and y-coordinates in the same order in the numerator and denominator when calculating slope. This will help avoid error.


Your Notes




x

y

1

3

12121

23 3

x

y

1

3

12121

23

23 3 5

Solution

a. m 5 y2 2 y1 } x2 2 x1


5 5 2

24 2 Substitute.

5 Simplify. The line is . The slope is .

b. m 5 y2 2 y1 } x2 2 x1


5 3 2

4 2 Substitute.

5 Simplify. The line is . The slope is .

Example 2 Find the slope of a line

Checkpoint Find the slope of the line passing through the points. Then classify the line by its slope.

3. (1, 22) and (1, 3) 4. (23, 7) and (4, 7)


Your Notes




x

y

1

3

12121

23 3

x

y

1

3

12121

23

23 3 5

Solution

a. m 5 y2 2 y1 } x2 2 x1


5

55 2

224 2 Substitute.

5 0 } 26 Simplify.

The line is horizontal . The slope is zero .

b. m 5 y2 2 y1 } x2 2 x1


5

(22)3 2

4 4 2 Substitute.

5 5 } 0 Simplify. The line is vertical . The slope is undefined .

Example 2 Find the slope of a line

Checkpoint Find the slope of the line passing through the points. Then classify the line by its slope.

3. (1, 22) and (1, 3) 4. (23, 7) and (4, 7)

undefined; vertical 0; horizontal


Your Notes


Gas Prices The table shows the cost of a gallon of gas for a number of days. Find the rate of change with respect to time.

Time (days) Day 1 Day 3 Day 5

Price/gal ($) 1.99 2.09 2.19

Rate of change 5 change in cost

}} change in time Write formula.

5 2.09 2

3 2 Substitute.

5 5 Simplify.

The rate of change in price is per day.

Example 3 Find a rate of change

5. The table shows the change in temperature over time. Find the rate of change in degrees Fahrenheit with respect to time.

Temperature (8F) Time (hours)

38 0

43 2

48 4

53 6

Checkpoint

Homework


Your Notes


Gas Prices The table shows the cost of a gallon of gas for a number of days. Find the rate of change with respect to time.

Time (days) Day 1 Day 3 Day 5

Price/gal ($) 1.99 2.09 2.19

Rate of change 5 change in cost

}} change in time Write formula.

5 1.992.09 2

13 2 Substitute.

5 2

0.1 5 0.05 } 1 Simplify.

The rate of change in price is 5 cents per day.

Example 3 Find a rate of change

5. The table shows the change in temperature over time. Find the rate of change in degrees Fahrenheit with respect to time.

Temperature (8F) Time (hours)

38 0

43 2

48 4

53 6

2.58F per hour

Checkpoint

Homework


Your Notes


4.5 Graph Using Slope-Intercept FormGoal p Graph linear equations using slope-intercept form.

VOCABULARY

Slope-intercept form

Parallel


FINDING THE SLOPE AND Y-INTERCEPT OF A LINE

Words Symbols

A linear equation of y 5 mx 1 bthe form y 5 mx 1 b is written in

where is the slope and is the y-intercept of the equation's graph. y 5 2x 1 1

Graph

x

y

1

3

12121

23

23 3 5

Your Notes


4.5 Graph Using Slope-Intercept FormGoal p Graph linear equations using slope-intercept form.

VOCABULARY

Slope-intercept form A linear equation written in the form y 5 mx 1 b

Parallel Two lines in the same plane that do not intersect


FINDING THE SLOPE AND Y-INTERCEPT OF A LINE

Words Symbols

A linear equation of y 5 mx 1 bthe form y 5 mx 1 b is written in slope-intercept form slope y-intercept where m is the slope and b is the y-intercept of the equation's graph. y 5 2x 1 1

Graph

x

y

1

3

12121

23

23 3 5

2

1 y 5 2x 1 1

Your Notes


Identify the slope and y-intercept of the line with the given equation.

a. y 5 x 1 3 b. 22x 1 y 5 5

Solution

a. The equation is in the form . So, the slope of the line is , and the y-intercept is .

b. Rewrite the equation in slope-intercept form by solving for .

22x 1 y 5 5 Write original equation.

y 5 Subtract from each side.

The line has a slope of and a y-intercept of .

Example 1 Identify slope and y-intercept

Checkpoint Identify the slope and y-intercept of the line with the given equation.

1. y 5 4x 2 1 2. 4x 2 2y 5 8

3. 4y 5 3x 1 16 4. 6x 1 3y 5 221


Your Notes


Identify the slope and y-intercept of the line with the given equation.

a. y 5 x 1 3 b. 22x 1 y 5 5

Solution

a. The equation is in the form y 5 mx 1 b . So, the slope of the line is 1 , and the y-intercept is 3 .

b. Rewrite the equation in slope-intercept form by solving for y .

22x 1 y 5 5 Write original equation.

y 5 2x 1 5 Subtract 22x from each side.

The line has a slope of 2 and a y-intercept of 5 .

Example 1 Identify slope and y-intercept

Checkpoint Identify the slope and y-intercept of the line with the given equation.

1. y 5 4x 2 1 2. 4x 2 2y 5 8

slope: 4 slope: 2

y-intercept: 21 y-intercept: 24

3. 4y 5 3x 1 16 4. 6x 1 3y 5 221

slope: 3 } 4 slope: 22

y-intercept: 4 y-intercept: 27


Your Notes



5. Graph the equation 2 1 } 2 x 1 y 5 1.

x

y

1

3

12121

23

23 3


Graph the equation 4x 1 y 5 2.

SolutionStep 1 Rewrite the equation in slope-intercept form.

Step 2 the slope and the y-intercept.

m 5 b 5

Step 3 the point that corresponds to the y-intercept, ( ).

Step 4 Use the slope to locate a second point on the line. Draw a line through the two points.

x

y

1

3

12121

23

23 3

Example 2 Graph an equation using slope-intercept formYour Notes



5. Graph the equation 2 1 } 2 x 1 y 5 1.

m 5 1 } 2

x

y

1

3

12121

23

23 3

21

(0, 1)(2, 2) b 5 1


Graph the equation 4x 1 y 5 2.

SolutionStep 1 Rewrite the equation in slope-intercept form.

y 5 24x 1 2

Step 2 Identify the slope and the y-intercept.

m 5 –4 b 5 2

Step 3 Plot the point that corresponds to the y-intercept, ( 0, 2 ).

Step 4 Use the slope to locate a second point on the line. Draw a line through the two points.

x

y

1

3

12121

23

23 3

1

24

Example 2 Graph an equation using slope-intercept formYour Notes


Homework

Determine which of the lines are parallel.

x

y

1

3

12121

25

23 5

Line a

Line c

Line b

Solution

Find the slope of each line.

Line a: m 5 2 3

2 4 5 5

Line b: m 5 2 4

2 2 5 5

Line c: m 5 2 2

2 6 5 5

Lines and have the same slope. They are parallel.

Example 3 Identify parallel lines

6. Determine which lines are parallel.

Line a: through (2, 5) and (22, 2)

Line b: through (4, 1) and (23, 24)

Line c: through (2, 3) and (22, 0)



Your Notes


Homework

Determine which of the lines are parallel.

x

y

1

3

12121

25

23 5

Line a

Line c

Line b

Solution

Find the slope of each line.

Line a: m 5

23 2 3

22 2 4 5

26

26 5 1

Line b: m 5

22 2 4

24 2 2 5

26

26 5 1

Line c: m 5

25 2 2

23 2 6 5

29

27 5

7 } 9

Lines a and b have the same slope. They are parallel.

Example 3 Identify parallel lines

6. Determine which lines are parallel.

Line a: through (2, 5) and (22, 2)

Line b: through (4, 1) and (23, 24)

Line c: through (2, 3) and (22, 0)

Lines a and c are parallel with slope 3 } 4 .



Your Notes



Solve Linear Equations by Graphing Goal p Use graphs to solve linear equations.

SOLVING LINEAR EQUATIONS GRAPHICALLY

Step 1 Write the equation in the form ax 1 b 5 0.

Step 2 Write the related function .

Step 3 Graph the equation .

The solution of ax 1 b 5 0 is the of the graph of .

Solve 4 } 3 x 1 1 5 2x graphically. Check your solution

algebraically.

Solution

Step 1 Write the equation in the form 1 5 0.

4 } 3 x 1 1 5 2x Write original equation.

1 5 0 Subtract from each side.

Step 2 Write the y 5 .

Step 3 Graph y 5 .

The is .

The solution of 4 } 3 x 1 1 5 2x is .

CHECK

4 } 3 ( ) 1 1 5 2( ) Substitute.

1 1 5 Simplify.

5 Solution checks.

Example 1 Solve an equation graphically


y

x1

1

Your Notes



Solve Linear Equations by Graphing Goal p Use graphs to solve linear equations.

SOLVING LINEAR EQUATIONS GRAPHICALLY


Step 2 Write the related function y 5 ax 1 b .

Step 3 Graph the equation y 5 ax 1 b .

The solution of ax 1 b 5 0 is the x-intercept of the graph of y 5 ax 1 b .

Solve 4 } 3 x 1 1 5 2x graphically. Check your solution

algebraically.

Solution


4 } 3 x 1 1 5 2x Write original equation.

2 2 } 3 x 1 1 5 0 Subtract 2x from each side.

Step 2 Write the related function y 5 2 2 } 3 x 1 1.

Step 3 Graph y 5 2 2 } 3 x 1 1.

The x-intercept is 3 } 2 .

The solution of 4 } 3 x 1 1 5 2x is 3 } 2 .

CHECK

4 } 3 ( 3 } 2 ) 1 1 5 2( 3 } 2 ) Substitute.

2 1 1 5 3 Simplify.

3 5 3 Solution checks.

Example 1 Solve an equation graphically


y

x1

1

Your Notes


1. Solve the equation graphically. Check your solution. 7x − 3 5 4x

2. Internet The number s (in millions) of subscribers to an Internet service can be modeled by the function s 5 1.85t 1 63.4 where t is the number of years since 1997. Use a graphing calculator to approximate the year when the number of subscribers will be 100 million.

The number c (in millions) of cell phone subscribers can be modeled by the function c 5 2.79t 1 45.1 where t is the number of years since 1999. In approximately what year was the total number of users 50 million?

Solution

Substitute for c in the linear model. The resulting linear equation is 5 1 .

Step 1 Write the equation in the form 1 5 0.

0 5 Subtract from each side.

0 5 Substitute for .

Step 2 Write the related function: y 5 .

Step 3 Graph the related function on a graphing calculator. Draw the line you obtain. The x-intercept is closest to .

Because x is the number of years since , you can estimate that the number

of cell phone users reached 50 million in about years, or in .

Example 2 Approximate a real-world solution



Use this viewing window:Xmin 5 –1Xmax 5 3Xsci 5 1Ymin 5 –7Ymax 5 3Ysci 5 1

y

x1

1

Homework

Your Notes


1. Solve the equation graphically. Check your solution. 7x − 3 5 4x

The solution is 1.

7(1) 2 3 5 4(1)

4 5 4

2. Internet The number s (in millions) of subscribers to an Internet service can be modeled by the function s 5 1.85t 1 63.4 where t is the number of years since 1997. Use a graphing calculator to approximate the year when the number of subscribers will be 100 million.

2017

The number c (in millions) of cell phone subscribers can be modeled by the function c 5 2.79t 1 45.1 where t is the number of years since 1999. In approximately what year was the total number of users 50 million?

Solution

Substitute 50 for c in the linear model. The resulting linear equation is 50 5 2.79t 1 45.1 .


0 5 2.79t 2 4.9 Subtract 50 from each side.

0 5 2.79x 2 4.9 Substitute x for t .

Step 2 Write the related function: y 5 2.79x 2 4.9 .

Step 3 Graph the related function on a graphing calculator. Draw the line you obtain. The x-intercept is closest to 2 .

Because x is the number of years since 1999 , you can estimate that the number of cell phone users reached 50 million in about 2 years, or in 2001 .

Example 2 Approximate a real-world solution


X�1.76 Y�0.01


Use this viewing window:Xmin 5 –1Xmax 5 3Xsci 5 1Ymin 5 –7Ymax 5 3Ysci 5 1

y

x1

1

Homework

Your Notes


4.6 Model Direct VariationGoal p Write and graph direct variation equations.

VOCABULARY

Direct variation

Constant of variation

Tell whether the equation represents direct variation. If so, identify the constant of variation.

a. 4x 1 2y 5 0 b. 22x 1 y 5 3

SolutionTo tell whether an equation represents direct variation, try to rewrite the equation in the form y 5 ax.

a. 4x 1 2y 5 0 Write original equation.

2y 5 Subtract from each side.

y 5 Simplify.

Because the equation 4x 1 2y 5 0 be rewritten in the form y 5 ax, it direct variation. The constant of variation is .

b. 22x 1 y 5 3 Write original equation.

y 5 1 3 Add to each side.

Because the equation 22x 1 y 5 3 be rewritten in the form y 5 ax, it direct variation.

Example 1 Identify direct variation equations


Your Notes


4.6 Model Direct VariationGoal p Write and graph direct variation equations.

VOCABULARY

Direct variation The relationship of two variables, x and y, provided y 5 ax and a Þ 0

Constant of variation In y 5 ax, a is called the constant of variation.

Tell whether the equation represents direct variation. If so, identify the constant of variation.

a. 4x 1 2y 5 0 b. 22x 1 y 5 3

SolutionTo tell whether an equation represents direct variation, try to rewrite the equation in the form y 5 ax.

a. 4x 1 2y 5 0 Write original equation.

2y 5 24x Subtract 4x from each side.

y 5 22x Simplify.

Because the equation 4x 1 2y 5 0 can be rewritten in the form y 5 ax, it represents direct variation. The constant of variation is 22 .

b. 22x 1 y 5 3 Write original equation.

y 5 2x 1 3 Add 2x to each side.

Because the equation 22x 1 y 5 3 cannot be rewritten in the form y 5 ax, it does not represent direct variation.

Example 1 Identify direct variation equations


Your Notes


Checkpoint Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. 3x 1 4y 5 0 2. 5x 1 y 5 1

Graph the direct variation equation.

a. y 5 25x b. y 5 3 } 5 x

Solutiona. Plot a point at the origin. The slope is equal to the

constant of variation, or . Find and plot a second point, then draw a line through the points.

x

y

1

12121

23

25

23 3

b. Plot a point at the origin. The slope is equal to the

constant of variation, or . Find and plot a second point, then draw a line through the points.

x

y

1

3

5

12121

23 3 5

Example 2 Graph direct variation equations

The graph of a direct variation equation is a line with a slope of a and a y-intercept of 0. This line passes through the origin.


Your Notes


Checkpoint Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. 3x 1 4y 5 0 2. 5x 1 y 5 1

yes; 2 3 } 4 no

Graph the direct variation equation.

a. y 5 25x b. y 5 3 } 5 x

Solutiona. Plot a point at the origin. The slope is equal to the

constant of variation, or 25 . Find and plot a second point, then draw a line through the points.

x

y

1

12121

23

25

23 3

1

25

(0, 0)

(1, 25)

b. Plot a point at the origin. The slope is equal to the

constant of variation, or 3 } 5 . Find and plot a second

point, then draw a line through the points.

x

y

1

3

5

12121

23 3 5

5

3

(0, 0)

(5, 3)

Example 2 Graph direct variation equations

The graph of a direct variation equation is a line with a slope of a and a y-intercept of 0. This line passes through the origin.


Your Notes


Homework

Your Notes

The graph of a direct variation

x

y3

121

23

25

23 3

(0, 0)

(24, 23)

equation is shown.

a. Write the direct variation equation.

b. Find the value of y when x 5 80.

Solutiona. Because y varies directly with x, the equation has the

form y 5 ax. Use the fact that y 5 23 when x 5 24 to find a.

y 5 ax Write direct variation equation.

5 a( ) Substitute.

5 a Solve for a. A direct variation equation that relates x and y is

y 5 .

b. When x 5 80, y 5 5 .

Example 3 Write and use a direct variation equation

3. Graph the direct variation equation y 5 1 } 2 x.

x

y

1

3

12121

23 3

4. The graph of a direct variation equation passes through the point (3, 24). Write the direct variation equation and find the value of y when x 5 15.




Homework

Your Notes

The graph of a direct variation

x

y3

121

23

25

23 3

(0, 0)

(24, 23)

equation is shown.

a. Write the direct variation equation.

b. Find the value of y when x 5 80.

Solutiona. Because y varies directly with x, the equation has the

form y 5 ax. Use the fact that y 5 23 when x 5 24 to find a.

y 5 ax Write direct variation equation.

23 5 a( 24 ) Substitute.

3 } 4 5 a Solve for a.

A direct variation equation that relates x and y is

y 5 3 } 4 x .

b. When x 5 80, y 5 3 } 4 (80) 5 60 .

Example 3 Write and use a direct variation equation

3. Graph the direct variation equation y 5 1 } 2 x.

x

y

1

3

12121

23 3

21

(0, 0)

(2, 1)

4. The graph of a direct variation equation passes through the point (3, 24). Write the direct variation equation and find the value of y when x 5 15.

y 5 2 4 } 3 x; y 5 220




4.7 Graph Linear FunctionsGoal p Use function notation.

VOCABULARY

Function notation

Family of functions

Parent linear function

For the function f(x) 5 3x 1 1, find the value of x so that f(x) 5 10.

Solution f(x) 5 3x 1 1 Write original equation.

5 3x 1 1 Substitute for f(x).

5 x Solve for x.

When x 5 , f(x) 5 10.

Example 1 Find an x-value

1. For f(x) 5 6x 2 6, find the value of x so that f(x) 5 24.



Your Notes



4.7 Graph Linear FunctionsGoal p Use function notation.

VOCABULARY

Function notation A way to name a function that is defined by an equation; f(x) 5 mx 1 b

Family of functions A group of functions with similar characteristics

Parent linear function The most basic linear function in a family of linear equations: f (x) 5 x

For the function f(x) 5 3x 1 1, find the value of x so that f(x) 5 10.

Solution f(x) 5 3x 1 1 Write original equation.

10 5 3x 1 1 Substitute 10 for f(x).

3 5 x Solve for x.

When x 5 3 , f(x) 5 10.

Example 1 Find an x-value


x 5 5


x 5 2


Your Notes



Text Messages A wireless communication provider estimates that the number of text messages m (in millions) sent over several years can be modeled by the function m 5 120t 1 95 where t represents the number of years since 2002. Graph the function and identify its domain and range.

t m

0

1

2

3

t

m

10 2 3 4 5

1000

200

300

400

500

600

Years since 2002

Text

messag

es (

millio

ns)

The domain of the function is t ≥ . From the graph or table, you can see that the range of the function is m ≥ .


3. Use the model from Example 2 to find the value of t so that m 5 1055. Explain what the solution means in this situation.


PARENT FUNCTION FOR LINEAR FUNCTIONS

1. The is the most basic linear function.

2. is the form of the parent linear function.


Your Notes


Text Messages A wireless communication provider estimates that the number of text messages m (in millions) sent over several years can be modeled by the function m 5 120t 1 95 where t represents the number of years since 2002. Graph the function and identify its domain and range.

t m

0 95

1 215

2 335

3 455

t

m

10 2 3 4 5

1000

200

300

400

500

600

Years since 2002

Text

messag

es (

millio

ns)

The domain of the function is t ≥ 0 . From the graph or table, you can see that the range of the function is m ≥ 95 .


3. Use the model from Example 2 to find the value of t so that m 5 1055. Explain what the solution means in this situation.

8; There will be over one billion text messages in 2010.


PARENT FUNCTION FOR LINEAR FUNCTIONS

1. The parent linear function is the most basic linear function.

2. f (x) 5 x is the form of the parent linear function.


Your Notes


Graph the function. Compare the graph with the graph of f(x) 5 x.

a. p(x) 5 x 2 4 b. q(x) 5 22x

Solutiona.

x

y

1

121

23

23 3 5

p(x) 5 x 2 4

b.

x

y

3

121

23

23 3

q(x) 5 22x

Because the graphs of Because the slope of p and f have the same the graph of q slope, m 5 1, the lines from left to right and are . Also, the slope of the graphthe y-intercept of the of f from left graph of p is less to right, the slope of than the y-intercept of q is . Thethe graph of f. y-intercept of both graphs is .

Example 3 Compare graphs with the graph f(x) 5 x

4. Graph r(x) 5 1 } 2 x. Compare the graph with the graph

of f (x) 5 x.

x

y

3

1

1

23

23 3



Your Notes


Graph the function. Compare the graph with the graph of f(x) 5 x.

a. p(x) 5 x 2 4 b. q(x) 5 22x

Solutiona.

x

y

1

121

23

23 3 5

p(x) 5 x 2 4

b.

x

y

3

121

23

23 3

q(x) 5 22x

Because the graphs of Because the slope of p and f have the same the graph of q falls slope, m 5 1, the lines from left to right and are parallel . Also, the slope of the graphthe y-intercept of the of f rises from left graph of p is 4 less to right, the slope of than the y-intercept of q is negative . Thethe graph of f. y-intercept of both graphs is 0 .

Example 3 Compare graphs with the graph f(x) 5 x

4. Graph r(x) 5 1 } 2 x. Compare the graph with the graph

of f (x) 5 x.

x

y

3

1

1

23

23 3

r (x) 5 x12

The slope of the graph of r is less than the slope of f. The y-intercept for both graphs is 0.



Your Notes


Homework


COMPARING GRAPHS OF LINEAR FUNCTIONS WITH THE GRAPH OF f(x) 5 x

g(x) 5 x 1 b The graphs have the same

x

y

3

1

121

23

23 3

.

The graphs have different .

Graphs of this family are of

the graph of f(x) 5 x.

g(x) 5 mx where m > 0 The graphs have different

x

y

3

1

1

23

23 3

(positive) .

The graphs have the same .

Graphs of this family are vertical or of the graph of f(x) 5 x.

g(x) 5 mx where m < 0 The graphs have different

x

y

3

121

23

23 3

(negative) .

The graphs have the same .

Graphs of this family are vertical or or of the graph of f(x) 5 x.

Your Notes


Homework


COMPARING GRAPHS OF LINEAR FUNCTIONS WITH THE GRAPH OF f(x) 5 x

g(x) 5 x 1 b The graphs have the same

x

y

3

1

121

23

23 3

slope .

The graphs have different y-intercepts .

Graphs of this family are vertical translations of

the graph of f(x) 5 x.

g(x) 5 mx where m > 0 The graphs have different

x

y

3

1

1

23

23 3

(positive) slopes .

The graphs have the same y-intercept .

Graphs of this family are vertical stretches or shrinks of the graph of f(x) 5 x.

g(x) 5 mx where m < 0 The graphs have different

x

y

3

121

23

23 3

(negative) slopes .

The graphs have the same y-intercept .

Graphs of this family are vertical stretches or shrinks or reflections of the graph of f(x) 5 x.

Your Notes


Words to Review

116 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Quadrant Transformation

Translation Vertical stretch

Vertical shrink Reflection

Give an example of the vocabulary word.


Words to Review


Quadrant

x

y

1

12123 321

Quadrant I

Quadrant IV

Quadrant II

Quadrant III

Transformation

(x, y) → (image x, image y)

Translation

(x, y) → (x 2 2, y)

y

x1

1

Vertical stretch

(x, y) → (x, 2y)

y

x

1

1

Vertical shrink

(x, y) → (x, 1 } 3 y)

y

x

11

Reflection

(x, y) → (x, –y)y

x1

1

Give an example of the vocabulary word.


Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 117

Solution of an equation in two variables.


Linear equation Standard form of a linear equation

Linear function Discrete function

Continuous function x-intercept

y-intercept Slope


Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 117

Solution of an equation in two variables.

x 1 y 5 2

(2, 0)


x

y

1

3

12123 321

23

(2, 1)

(1, 21)

(21, 25)

Linear equation

2x 1 4y 5 8

Standard form of a linear equation

Ax 1 By 5 C

Linear function

y 5 x 1 1

Discrete functiony

x1

1

Continuous functiony

x1

1

x-intercept

The x-intercept of 2x 1 3y 5 18 is 9.

y-intercept

The y-intercept of 2x 1 3y 5 18 is 6.

Slope

slope 5 rise } run

5 change in y

} change in x


Review your notes and Chapter 4 by using the Chapter Review on pages 276–279 of your textbook.

Rate of change Slope-intercept form

Parallel

Direct variation

Constant of variation Function notation

Family of functions Parent linear function



Review your notes and Chapter 4 by using the Chapter Review on pages 276–279 of your textbook.

Rate of change

change in cost

}} change in time

Slope-intercept form

y 5 mx 1 b

Parallel

Lines in the same plane that do not intersect

Direct variation

22x 1 y 5 0

y 5 2x

Constant of variation

y 5 2x

constant of variation 5 2

Function notation

f (x) 5 mx 1 b

Family of functions

Of the form:

f (x) 5 mx 1 b

family of linear functions

Parent linear function

f (x) 5 x



4.1 plot points in a coordinate plane - rjs solutions · lah_a1_11_fl_ntg_ch04_85-118.indd 89...

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