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Copyright © by Holt, Rinehart and Winston. 22 Holt Algebra 2 All rights reserved. Use intercepts to sketch the graph of the function 3x 6y 12. The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x. 3x 6y 12 3x 6(0) 12 3x 12 x 4 The y-intercept is where the graph crosses the y-axis. To find the y-intercept, set x 0 and solve for y. 3x 6y 12 3(0) 6y 12 6y 12 y 2 Plot the points ( 4, 0) and ( 0, 2). Draw a line connecting the points. Find the intercepts and graph each line. Name Date Class Reteach Graphing Linear Functions 2-3 LESSON 1. 3x 2y 6 a. 3x 2 6 x-intercept b. 3 2y 6 y-intercept 2. 6x 3y 12 a. 6x 3 12 x-intercept b. 6 3y 12 y-intercept The y-intercept occurs at the point 0, 2 . The x-intercept occurs at the point 4, 0 .

LESSON Reteach Graphing Linear Functions - Klein …classroom.kleinisd.net/users/0262/docs/review_for_unit_… ·  · 2008-02-06... set x 0 and solve for y. 3x 6y 12 3(0) 6y 12 6y

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Copyright © by Holt, Rinehart and Winston. 22 Holt Algebra 2All rights reserved.

Use intercepts to sketch the graph of the function 3x 6y 12.

The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x.

3x 6y 12

3x 6(0) 12

3x 12

x 4

The y-intercept is where the graph crosses the y-axis. To find the y-intercept, set x 0 and solve for y.

3x 6y 12

3(0) 6y 12

6y 12

y 2

Plot the points ( 4, 0) and ( 0, 2). Draw a line connecting the points.

Find the intercepts and graph each line.

Name Date Class

ReteachGraphing Linear Functions2-3

LESSON

1. 3x 2y 6

a. 3x 2 6

x-intercept

b. 3 2y 6

y-intercept

2. 6x 3y 12

a. 6x 3 12

x-intercept

b. 6 3y 12

y-intercept

The y-intercept occurs at the point 0, 2 .

The x-intercept occurs at the point 4, 0 .

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a207c02-3_rt.indd 22 12/7/05 6:28:36 PM

Copyright © by Holt, Rinehart and Winston. 23 Holt Algebra 2All rights reserved.

Write each function in slope-intercept form. Use m and b to graph.

ReteachGraphing Linear Functions (continued)

Name Date Class

2-3LESSON

y mx b is the slope-intercept form.m represents the slope and b represents the y-intercept.

Use the slope and the y-intercept to graph a linear function.

To write 2y x 6 in slope-intercept form, solve for y.

2y x 6

x x

2y x 6

2y

___ 2 x __

2 6 __

2

y 1 __ 2 x 3

Compare y 1 __ 2 x 3 to y mx b.

m 1 __ 2 , so the slope is 1 __

2 .

b 3, so the y-intercept is 3.

3. 2x y 1

a. y x

b. m

c. b

4. y x __ 2 1

a. y

b. m

c. b

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a207c02-3_rt.indd 23 12/7/05 6:28:37 PM

Copyright © by Holt, Rinehart and Winston. 30 Holt Algebra 2All rights reserved.

Write the equation of the line shown in the graph in slope-intercept form.

Slope-intercept form: y mx b

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The point 2, 4 lies on the line.

From 0, 1 , move 3 units down, or a rise of 3 units, and 2 units right, or a run of 2 units, to 2, 4 .

m rise ____ run 3 ___ 2 3 __

2

Note that when the rise is a drop the slope is negative.

Substitute m 3 __ 2 and b 1 into y mx b to get the equation y 3 __

2 x 1.

Write the equation of each line in slope-intercept form.

1.

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b b b

m rise ____ run 1 __ 1 m rise ____ run _____

1 m rise ____ run _____

y x y y

Name Date Class

ReteachWriting Linear Functions2-4

LESSON

a207c02-4_rt.indd 30 12/7/05 6:28:47 PM

Copyright © by Holt, Rinehart and Winston. 31 Holt Algebra 2All rights reserved.

Write the equation of each line.

4. parallel to y 4x 3 through the point 1, 2

m x 1 , y 1 ,

y y 1 m x x 1 → y x y

5. perpendicular to y 1 __ 2

x 4 through the point 1, 1

m x 1 , y 1 ,

y y 1 m x x 1 → y x y

The negative reciprocal of 1 __ 3

is

3 because 1 __ 3 3 1.

ReteachWriting Linear Functions (continued)

Name Date Class

2-4 LESSON

The slopes of parallel and perpendicular lines have a special relationship.

The slopes of parallel lines are equal.

y 2x 1 and y 2x 2 are parallel

lines since both equations have a slope of 2.

Note: The slopes of parallel vertical lines are

undefined.

The slopes of perpendicular lines are negative

reciprocals. Their product is 1.

y 2x 1 and y 1 __ 2 x 1 are perpendicular

since 2 1 __ 2

1.

The point-slope form of the equation of a line is y y 1 m x x 1 .

The line has slope m and passes through the point x 1 , y 1 .

Write the equation of the line perpendicular to y 1 __ 3

x 2 through 2, 5 .

Substitute values for m and x 1 , y 1 in y y 1 m x x 1 .

x 1 , y 1 2, 5 , so x 1 2, y 1 5, and m 3

y y 1 m x x 1 → y 5 3 x 2

y 5 3x 6

y 3x 11

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a207c02-4_rt.indd 31 12/7/05 6:28:48 PM

Copyright © by Holt, Rinehart and Winston. 46 Holt Algebra 2All rights reserved.

Name Date Class

ReteachTransforming Linear Functions2-6

LESSON

Translating linear functions vertically or horizontally changes the intercepts of the function. It does NOT change the slope.

Let f x 3x 1. Read the rule for each translation.

Horizontal Translation BThink: Add to x, go west.Use f x r f x h .

Vertical Translation n Think: Add to y, go high.Use f x r f x k.

Translation 2 units right c

g x f x 2 g x 3 x 2 1

Rule: g x 3x 5

Translation 2 units up M

g x f x 2g x 3x 1 2

Rule: g x 3x 3

Translation 2 units left V

h x f x 2 f x 2 h x 3 x 2 1

Rule: h x 3x 7

Translation 2 units down m

h x f x 2 f x 2h x 3x 1 2

Rule: h x 3x 1

Let f x 2x 1. Write the rule for g x .

1. horizontal translation 5 units right 2. vertical translation 4 units down

g x f x g x f x

g x 2 x 1 g x

g x g x

3. vertical translation 3 units up 4. horizontal translation 1 unit left

g x f x g x f x g x g x 2 1

g x g x

5. vertical translation 7 units down 6. horizontal translation 9 units right

7. vertical translation 1 unit up 8. horizontal translation 1 __ 2 unit to the left

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a207c02-6_rt.indd 46 12/7/05 6:29:03 PM

Copyright © by Holt, Rinehart and Winston. 47 Holt Algebra 2All rights reserved.

ReteachTransforming Linear Functions (continued)

Name Date Class

2-6LESSON

Compressing or stretching linear functions changes the slope.Let f � x � � 3x � 1. Read the rule for each translation.

Horizontal stretch or compression by a factor of b

Use f � x � r f � 1 __ b x � .

Vertical stretch or compression by a factor of a

Use f � x � r a � f � x � .

Horizontal stretch by a factor of 2

g � x � � f � 1 __ b x � � f � 1 __

2 x �

g � x � � 3 � 1 __ 2 x � � 1

Rule: g � x � � 3 __ 2 x � 1

Vertical stretch by a factor of 2

g � x � � a � f � x � � 2 � f � x �

g � x � � 2 � 3x � 1 �

Rule: g � x � � 6x � 2

Horizontal compression by a factor of 1 __ 2

h � x � � f � 1 __ b x � � f � 1 ___

� 1 __ 2 � x � � f � 2x �

h � x � � 3 � 2x � � 1

Rule: h � x � � 6x � 1

Vertical compression by a factor of 1 __ 2

h � x � � a � f � x � � 1 __ 2 � f � x �

h � x � � 1 __ 2 � 3x � 1 �

Rule: h � x � � 3 __ 2 x � 1 __

2

Let f � x � � 2x � 1. Write the rule for g � x � .

7. vertical compression by a factor of 1 __ 4 8. horizontal stretch by a factor of 3

f � x � r 1 __ 4

� f � x � f � x � r f � 1 __ 3 x

g � x � � 1 __ 4

� 2x � 1 � g � x � � 2 � 1 __ 3 x

� � 1

g � x � � 1 __ 2 x � 1 __

4 g � x � �

2 __ 3 x � 1

9. horizontal compression by a factor of 1 __ 3 10. vertical stretch by a factor of 5

f � x � r

2 � 1 ____ 1 __ 3 x � � 1

f � x � r 5 � 2x � 1 �

g � x � � 6x � 1 g � x � r g � x � � 10x � 5

a207c02-6_rt.indd 47a207c02-6_rt.indd 47 6/9/06 5:02:11 PM6/9/06 5:02:11 PMProcess BlackProcess Black

Copyright © by Holt, Rinehart and Winston. 54 Holt Algebra 2All rights reserved.

Name Date Class

ReteachCurve Fitting with Linear Models2-7

LESSON

Use a scatter plot to identify a correlation. If the variables appear correlated, then find a line of fit.

Positive correlation Negative correlation No correlation

The table shows the relationship between two variables. Identify the correlation, sketch a line of fit, and find its equation.

x 1 2 3 4 5 6 7 8

y 16 14 11 10 5 2 3 2

Step 1 Make a scatter plot of the data.As x increases, y decreases.The data is negatively correlated.

Step 2 Use a straightedge to draw a line.There will be some points above and below the line.

Step 3 Choose two points on the line to find the equation: � 1, 16 � and � 7, 2 � .

Step 4 Use the points to find the slope:

m � change in y

__________ change in x

� 16 � 2 ______ 1 � 7

� 14 ___ �6

� � 7 __ 3

Step 5 Use the point-slope form to find the equation of aline that models the data.

y � y 1 � m � x � x 1 �

y � 2 � � 7 __ 3 � x � 7 �

y � � 7 __ 3 x � 18

Use the scatter plot of the data to solve.

1. The correlation is Positive .

2. Choose two points on the line and find the slope.

� 4, 8 � and � 6, 11 � ; m � 3 __ 2

3. Find the equation of a line that models the data.

y � 3 __ 2 x � 2

a207c02-7_rt.indd Sec1:54a207c02-7_rt.indd Sec1:54 12/29/05 7:42:01 PM12/29/05 7:42:01 PMProcess BlackProcess Black

Copyright © by Holt, Rinehart and Winston. 55 Holt Algebra 2All rights reserved.

ReteachCurve Fitting with Linear Models (continued)

Name Date Class

2-7LESSON

A line of best fit can be used to predict data.

Use the correlation coefficient, r, to measure how well the data fits.

�1 � r � 1

Use a graphing calculator to find the correlation coefficient of the data and the line of best fit. Use STAT EDIT to enter the data.

x 1 2 3 4 5 6 7 8

y 16 14 11 10 5 2 3 2

Use LinReg from the STAT CALC menu to find the line of best fit and the correlation coefficient.

LinReg y � ax � b a � �2.202 b � 17.786 r 2 � .9308 r � �.9648

Use the linear regression model to predict y when x � 3.5.

y � �2.2x � 17.79

y � �2.2 � 3.5 � � 17.79

y � 10.09

Use a calculator and the scatter plot of the data to solve.

4. Find the correlation coefficient, r. 0.965

5. Find the equation of the line of best fit.

y � 1.38x � 2.29

6. Predict y when x � 2.6. y � 5.878

7. Predict y when x � 5.3. y � 9.604

If r is near 1, data is modeled by a line with a positive slope.

If r is near �1, data is modeled by a line with a negative slope.

If r is near 0, datahas no correlation.

The correlation coefficient is �0.9648. The data is very close to linear with a negative slope.

a207c02-7_rt.indd Sec1:55a207c02-7_rt.indd Sec1:55 12/29/05 7:42:03 PM12/29/05 7:42:03 PMProcess BlackProcess Black