LESSON Reteach Graphing Linear Functions - Klein...
Transcript of LESSON Reteach Graphing Linear Functions - Klein...
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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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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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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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
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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.
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