4.1 plot points in a coordinate plane - rjs solutions · lah_a1_11_fl_ntg_ch04_85-118.indd 89...
TRANSCRIPT
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
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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.
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 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)
Checkpoint Complete the following exercise.
Your Notes
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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
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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
86 Lesson 4.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.1 • Algebra 1 Notetaking Guide 87
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
Checkpoint Complete the following exercise.
Your Notes
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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 1 Make a table.
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 .
Example 3 Graph a function
Homework
Copyright © Holt McDougal. All rights reserved. Lesson 4.1 • Algebra 1 Notetaking Guide 87
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
Checkpoint Complete the following exercise.
Your Notes
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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
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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.
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
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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
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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
Copyright © Holt McDougal. All rights reserved. 4.1 Focus on Graphing • Algebra 1 Notetaking Guide 89
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
Checkpoint Complete the following exercises.
y
x
1
1
Homework
Your Notes
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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
Step 2 Make a table.
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.
90 Lesson 4.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
LAH_A1_11_FL_NTG_Ch04_85-118.indd 90LAH_A1_11_FL_NTG_Ch04_85-118.indd 90 3/2/09 4:22:08 PM3/2/09 4:22:08 PM
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
Step 2 Make a table.
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.
90 Lesson 4.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.2 • Algebra 1 Notetaking Guide 91
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.
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.2 • Algebra 1 Notetaking Guide 91
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.
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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 .
92 Lesson 4.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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) .
92 Lesson 4.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.2 • Algebra 1 Notetaking Guide 93
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
Checkpoint Complete the following exercise.
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.2 • Algebra 1 Notetaking Guide 93
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
Checkpoint Complete the following exercise.
Your Notes
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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
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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
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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.
Checkpoint Complete the following exercise.
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
Checkpoint Complete the following exercise.
Homework
Your Notes
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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
Checkpoint Complete the following exercise.
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.
Checkpoint Complete the following exercise.
Homework
Your Notes
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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.
8x 2 2y 5 32 Write originalequation.
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
96 Lesson 4.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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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 2y 5 32 Write originalequation.
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.
8x 2 2y 5 32 Write originalequation.
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
96 Lesson 4.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.3 • Algebra 1 Notetaking Guide 97
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.3 • Algebra 1 Notetaking Guide 97
Your Notes
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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
Checkpoint Complete the following exercises.
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
98 Lesson 4.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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
Checkpoint Complete the following exercises.
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
98 Lesson 4.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.4 • Algebra 1 Notetaking Guide 99
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.4 • Algebra 1 Notetaking Guide 99
Your Notes
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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
Write formula for slope.
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.
100 Lesson 4.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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
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
Write formula for slope.
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
Write formula for slope.
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.
100 Lesson 4.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Find the slope of the line shown.
a. Let (x1, y1) 5 (2, 5) b. Let (x1, y1) 5 (4, 22) and (x2, y2) 5 (24, 5). and (x2, y2) 5 (4, 3).
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
Write formula for slope.
5 5 2
24 2 Substitute.
5 Simplify. The line is . The slope is .
b. m 5 y2 2 y1 } x2 2 x1
Write formula for slope.
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)
Copyright © Holt McDougal. All rights reserved. Lesson 4.4 • Algebra 1 Notetaking Guide 101
Your Notes
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Find the slope of the line shown.
a. Let (x1, y1) 5 (2, 5) b. Let (x1, y1) 5 (4, 22) and (x2, y2) 5 (24, 5). and (x2, y2) 5 (4, 3).
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
Write formula for slope.
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
Write formula for slope.
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
Copyright © Holt McDougal. All rights reserved. Lesson 4.4 • Algebra 1 Notetaking Guide 101
Your Notes
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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 ($) 1.99 2.09 2.19
Rate of change 5 change in cost
}} change in time Write formula.
5 2.09 2
3 2 Substitute.
5 5 Simplify.
The rate of change in price is per day.
Example 3 Find a rate of change
5. The table shows the change in temperature over time. Find the rate of change in degrees Fahrenheit with respect to time.
Temperature (8F) Time (hours)
38 0
43 2
48 4
53 6
Checkpoint
Homework
102 Lesson 4.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Gas Prices The table shows the cost of a gallon of gas for a number of days. Find the rate of change with respect to time.
Time (days) Day 1 Day 3 Day 5
Price/gal ($) 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
102 Lesson 4.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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4.5 Graph Using Slope-Intercept FormGoal p Graph linear equations using slope-intercept form.
VOCABULARY
Slope-intercept form
Parallel
Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Algebra 1 Notetaking Guide 103
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
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Algebra 1 Notetaking Guide 103
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
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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
104 Lesson 4.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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
104 Lesson 4.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Algebra 1 Notetaking Guide 105
5. Graph the equation 2 1 } 2 x 1 y 5 1.
x
y
1
3
12121
23
23 3
Checkpoint Complete the following exercise.
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
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Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Algebra 1 Notetaking Guide 105
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
Checkpoint Complete the following exercise.
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
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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)
Checkpoint Complete the following exercise.
106 Lesson 4.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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 .
Checkpoint Complete the following exercise.
106 Lesson 4.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Copyright © Holt McDougal. All rights reserved. 4.5 Focus on Graphing • Algebra 1 Notetaking Guide 107
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
Focus On GraphingUse after Lesson 4.2
y
x1
1
Your Notes
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Copyright © Holt McDougal. All rights reserved. 4.5 Focus on Graphing • Algebra 1 Notetaking Guide 107
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 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
Step 1 Write the equation in the form ax 1 b 5 0.
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
Focus On GraphingUse after Lesson 4.2
y
x1
1
Your Notes
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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
Checkpoint Complete the following exercises.
108 4.5 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Use this viewing window:Xmin 5 –1Xmax 5 3Xsci 5 1Ymin 5 –7Ymax 5 3Ysci 5 1
y
x1
1
Homework
Your Notes
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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 .
Step 1 Write the equation in the form ax 1 b 5 0.
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
Checkpoint Complete the following exercises.
X�1.76 Y�0.01
108 4.5 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Use this viewing window:Xmin 5 –1Xmax 5 3Xsci 5 1Ymin 5 –7Ymax 5 3Ysci 5 1
y
x1
1
Homework
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.6 • Algebra 1 Notetaking Guide 109
Your Notes
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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
Copyright © Holt McDougal. All rights reserved. Lesson 4.6 • Algebra 1 Notetaking Guide 109
Your Notes
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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.
110 Lesson 4.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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.
110 Lesson 4.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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.
Checkpoint Complete the following exercises.
Copyright © Holt McDougal. All rights reserved. Lesson 4.6 • Algebra 1 Notetaking Guide 111
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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
Checkpoint Complete the following exercises.
Copyright © Holt McDougal. All rights reserved. Lesson 4.6 • Algebra 1 Notetaking Guide 111
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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.
2. For f(x) 5 7x 1 3, find the value of x so that f(x) 5 17.
Checkpoint Complete the following exercises.
Your Notes
112 Lesson 4.7 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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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
1. For f(x) 5 6x 2 6, find the value of x so that f(x) 5 24.
x 5 5
2. For f(x) 5 7x 1 3, find the value of x so that f(x) 5 17.
x 5 2
Checkpoint Complete the following exercises.
Your Notes
112 Lesson 4.7 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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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 ≥ .
Example 2 Graph a function
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.
Checkpoint Complete the following exercise.
PARENT FUNCTION FOR LINEAR FUNCTIONS
1. The is the most basic linear function.
2. is the form of the parent linear function.
Copyright © Holt McDougal. All rights reserved. Lesson 4.7 • Algebra 1 Notetaking Guide 113
Your Notes
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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 .
Example 2 Graph a function
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.
Checkpoint Complete the following exercise.
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.
Copyright © Holt McDougal. All rights reserved. Lesson 4.7 • Algebra 1 Notetaking Guide 113
Your Notes
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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
Checkpoint Complete the following exercise.
114 Lesson 4.7 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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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.
Checkpoint Complete the following exercise.
114 Lesson 4.7 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Homework
Copyright © Holt McDougal. All rights reserved. Lesson 4.7 • Algebra 1 Notetaking Guide 115
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
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Homework
Copyright © Holt McDougal. All rights reserved. Lesson 4.7 • Algebra 1 Notetaking Guide 115
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
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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.
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Words to Review
116 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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.
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Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 117
Solution of an equation in two variables.
Graph 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
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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)
Graph of an equation in two variables
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
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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
118 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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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
118 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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