Download Chapter 5

Name ———————————————————————
LESSON
5.1
Date ————————————
Study Guide
For use with pages 282–291
GOAL
EXAMPLE 1
Write equations of lines.
Use slope and y-intercept to write an equation
1
Write an equation of the line with a slope of }
and a y-intercept of 27.
2
Solution
y 5 mx 1 b
Write slope-intercept form.
1
1
y 5 }2 x 2 7
Substitute }2 for m and 27 for b.
Exercises for Example 1
1. slope: 7
y-intercept: 211
LESSON 5.1
Write an equation of the line with the given slope and y-intercept.
2
2. slope: }
3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y-intercept: 5
7
3. slope: 2}
5
y-intercept: 22
EXAMPLE 2
Write an equation of a line given two points
Write an equation of the line shown.
y
5
Solution
STEP 1
4
Calculate the slope.
y2 2 y1
2 2(21)
2
1
3
5}
5 2}2
m5}
x 2x
022
2
STEP 2
23 22
1
Write an equation of the line.
The line crosses the y-axis at (0, 2).
So, the y-intercept is 2.
y 5 mx 1 b
3
y 5 2}2 x 1 2
(0, 2)
O
22
1
3
4 5 x
(2, 21)
23
Write slope-intercept form.
3
Substitute 2}2 for m and 2 for b.
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Study Guide
Date ————————————
continued
For use with pages 282–291
Exercises for Example 2
Write an equation of the line that passes through the given points.
4. (10, 4), (0, 21)
5. (0, 8), (5, 21)
6. (26, 28), (0, 214)
LESSON 5.1
EXAMPLE 3
Write a linear function
Write an equation for the linear function f with the values f(0) 5 7 and
f(12) 5 15.
Solution
STEP 1
Write f (0) 5 7 as (0, 7) and f (12) 5 15 as (12, 15).
STEP 2
Calculate the slope of the line that passes through (0, 7) and (12, 15).
y2 2 y1
15 2 7
8
2
5}
5}
5 }3
m5}
x 2x
12 2 0
12
2
Write an equation of the line. The line crosses the y-axis at (0, 7). So, the
y-intercept is 7.
y 5 mx 1 b
Write slope-intercept form.
2
2
y 5 }3 x 1 7
Substitute }3 for m and 7 for b.
2
The function is f (x) 5 }3 x 1 7.
Exercises for Example 3
Write an equation for the linear function f with the given values.
7. f (0) 5 21, f (4) 5 13
8. f (3) 5 212, f (0) 5 6
10
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STEP 3
1
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LESSON
5.2
Date ————————————
Study Guide
For use with pages 292–299
GOAL
EXAMPLE 1
Write an equation of a line using points on the line.
Write an equation given the slope and a point
Write an equation of the line that passes through the point (2, 5) and
has a slope of 3.
Solution
STEP 1
Identify the slope. The slope is 3.
STEP 2
Find the y-intercept. Substitute the slope and the coordinates of the given
point into y 5 mx 1 b. Solve for b.
y 5 mx 1 b
Write slope-intercept form.
5 5 3(2) 1 b
Substitute 3 for m, 2 for x, and 5 for y.
21 5 b
EXAMPLE 2
Write an equation of the line.
y 5 mx 1 b
Write slope-intercept form.
y 5 3x 2 1
Substitute 3 for m, and 21 for b.
Write an equation given two points
Write an equation of the line that passes through (3, 9) and (22, 21).
Solution
STEP 1 Calculate the slope.
y2 2 y1
21 2 9
LESSON 5.2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
STEP 3
Solve for b.
210
m5}
5}
5}
52
25
x 2x
22 2 3
2
1
STEP 2 Find the y-intercept. Use the slope and the point (3, 9).
STEP 3
y 5 mx 1 b
Write slope-intercept form.
9 5 2(3) 1 b
Substitute 2 for m, 3 for x, and 9 for y.
35b
Solve for b.
Write an equation of the line.
y 5 mx 1 b
Write slope-intercept form.
y 5 2x 1 3
Substitute 2 for m and 3 for b.
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LESSON
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Study Guide
Date ————————————
continued
For use with pages 292–299
Exercises for Examples 1 and 2
Write an equation of the line that passes through the given point and
has the given slope.
1. (7, 2); m 5 4
1
2. (9, 15); m 5 2}
3
Write an equation of the line that passes through the two given points.
3. (5, 8), (13, 12)
4. (26, 27), (23, 5)
EXAMPLE 3
Write a linear function
Write an equation of the linear function with the values f (2) 5 3 and
f(23) 5 8.
Solution
STEP 1 Calculate the slope. Write f(2) 5 3 as (2, 3) and f (23) 5 8 as (23, 8).
y2 2 y1
823
5
m5}
5}
5}
5 21
25
x 2x
23 2 2
1
LESSON 5.2
STEP 2 Find the y-intercept. Use the slope and the point (2, 3).
STEP 3
y 5 mx 1 b
Write slope-intercept form.
3 5 21(2) 1 b
Substitute 21 for m, 2 for x, and 3 for y.
55b
Solve for b.
Write an equation for the function. Use f (x) 5 mx 1 b.
f (x) 5 2x 1 5
Substitute 21 for m and 5 for b.
Exercises for Example 3
Write an equation for a linear function f that has the given values.
5. f (2) 5 24 and f(24) 5 27
6. f (25) 5 17 and f (3) 5 9
20
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LESSON
5.3
Date ————————————
Study Guide
For use with pages 302–308
GOAL
Write linear equations in point-slope form.
Vocabulary
The point-slope form of the equation of the nonvertical line through a
given point (x1, y1) with a slope of m is y 2 y1 5 m(x 2 x1).
EXAMPLE 1
Write an equation in point-slope form
Write an equation in point-slope form of the line that passes through
the point (5, 1) and has a slope of 23.
Solution
y 2 y1 5 m(x 2 x1)
Write point-slope form.
y 2 1 5 23(x 2 5)
Substitute 23 for m, 5 for x, and 1 for y.
Exercises for Example 1
Write an equation in point-slope form of the line that passes through
the given point and has the given slope.
2. (1, 4); m 5 24
4
3. (6, 28); m 5 2}EXAMP
9
EXAMPLE 2
Graph an equation in point-slope form
4
Graph the equation y 2 2 5 2}
(x 1 4).
7
Solution
y
4
Because the equation is in point-slope
form, you know that the line has a
(24, 2)
4
7
slope of 2} and passes through the
point (24, 2). Plot the point (24, 2).
Find a second point on the line using
the slope. Draw a line through the
two points.
2
1
24
25
3
23 22
3 x
O
1 2
7
(3, ⫺2)
LESSON 5.3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1. (23, 22); m 5 5
24
Exercise for Example 2
4. Graph the equation y 1 3 5 4(x 1 2).
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LESSON
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Study Guide
Date ————————————
continued
For use with pages 302–308
EXAMPLE 3
Use point slope form to write an equation
Write an equation in point-slope form of
the line shown.
4
Solution
2
y
3
(3, 3)
1
STEP 1
Find the slope of the line.
23 22
O
y2 2 y1
22
23
m5}
x 2x
2
1
1
2
3
4
5 x
(1, 22)
24
3 2 (22)
5}
321
5
5 }2
Write the equation in point-slope form. You can use either point.
Method 1 Use (3, 3).
y 2 y1 5 m(x 2 x1)
y 2 y1 5 m(x 2 x1)
5
y 1 2 5 }2(x 2 1)
y 2 3 5 }2 (x 2 3)
CHECK
Method 2 Use (1, 22).
5
Check that the equations are equivalent by writing them in slope-intercept
form.
5
y 2 3 5 }2(x 2 3)
5
9
y 5 }2 x 2 }2
5
y 1 2 5 }2(x 2 1)
5
9
y 5 }2 x 2 }2
Exercises for Example 3
5. Write an equation in point-slope form of the line that passes through the points
(23, 8) and (4, 213).
6. Write an equation in point-slope form of the line that passes through the points
LESSON 5.3
(10, 26), (26, 8).
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STEP 2
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LESSON
LESSON 5.4
5.4
Date ————————————
Study Guide
For use with pages 311–316
GOAL
EXAMPLE 1
Write equations in standard form.
Write equivalent equations in standard form
Write two equations in standard form that are equivalent to 3x 2 9y 5 12.
Solution
To write another equivalent equation,
multiply each side by 2.
To write one equivalent equation,
1
multiply each side by }3.
x 2 3y 5 4
EXAMPLE 2
6x 2 18y 5 24
Write an equation from a graph
Write an equation in standard form of the
line shown.
4
Solution
2
y
3
(1, 3)
1
Calculate the slope.
25 24 23 22
y2 2 y1
313
6
m5}
5 }3 5 2
x2 2 x1 5 }
112
STEP 2
STEP 3
(22, 23)
Write an equation in point-slope form.
Use (1, 3).
O
1 2
3 x
23
24
y 2 y1 5 m(x 2 x1)
Write point-slope form.
y 2 3 5 2(x 2 1)
Substitute 2 for m, 1 for x, and 3 for y.
Rewrite the equation in standard form.
y 2 3 5 2x 2 2
22x 1 y 5 1
Distributive property
Collect variable terms on one side, constants
on the other.
Exercises for Examples 1 and 2
1. Write two equations in standard form that are equivalent to 6x 1 2y 5 8.
Write an equation in standard form of the line that passes through the
given points.
2. (4, 4), (8, 2)
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3. (22, 3), (24, 25)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
STEP 1
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LESSON
5.4
Study Guide
Date ————————————
continued
For use with pages 311–316
Write an equation of a line
Write equations of the horizontal and vertical lines that pass through
the point (22, 8).
Solution
LESSON 5.4
EXAMPLE 3
The horizontal line has all the same y-coordinates. The y-coordinate of the given point
is 8. So, an equation of the horizontal line is y 5 8.
The vertical line has all the same x-coordinates. The x-coordinate of the given point is
22. So, an equation of the vertical line is x 5 22.
Exercises for Example 3
Write equations of the horizontal and vertical lines that pass through
the given point.
4. (7, 22)
EXAMPLE 4
5. (21, 5)
Complete an equation in standard form
The graph of 5x 1 By 5 6 is a line that passes through the point (2, 1).
Find the missing coefficient and write the completed equation.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Solution
STEP 1
Find the value of B. Substitute the coordinates of the given point for x and y
in the equation. Solve for B.
5x 1 By 5 6
5(2) 1 B(1) 5 6
B 5 24
STEP 2
Write equation.
Substitute 2 for x and 1 for y.
Simplify.
Complete the equation.
5x 2 4y 5 6
Substitute 24 for B.
The completed equation is 5x 2 4y 5 6.
Exercises for Example 4
Find the missing coefficient in the equation of the line that passes
through the given point. Write the completed equation.
6. Ax 1 5y 5 7, (4, 3)
7. 4x 1 By 5 6, (3, 22)
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LESSON
5.5
Date ————————————
Study Guide
For use with pages 318–324
GOAL
Write equations of parallel and perpendicular lines.
Vocabulary
The converse of a conditional statement interchanges the hypothesis
and conclusion.
Parallel Lines
If two nonvertical lines have the same slope, then they are parallel.
If two nonvertical lines are parallel, then they have the same slope.
Perpendicular Lines
Two lines in a plane are perpendicular if they intersect to form a right
angle.
If two nonvertical lines are perpendicular, then their slopes are negative
reciprocals.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
EXAMPLE 1
LESSON 5.5
If two nonvertical lines have slopes that are negative reciprocals, then
the lines are perpendicular.
Write an equation of a parallel line
Write an equation of the line that passes through (2, 6) and is parallel to
the line y 5 2x 1 2.
Solution
STEP 1
Identify the slope. The graph of the given equation has a slope of 21. So, the
parallel line through (2, 6) has a slope of 21.
STEP 2
Find the y-intercept. Use the slope and the given point.
STEP 3
y 5 mx 1 b
Write slope-intercept form.
6 5 21(2) 1 b
Substitute 21 for m, 2 for x, and 6 for y.
85b
Solve for b.
Write the equation. Use y 5 mx 1 b.
y 5 2x 1 8
Substitute 21 for m and 8 for b.
Exercises for Example 1
Write an equation of the line that passes through the given point and is
parallel to the given line.
2
1. (9, 2), y 5 } x 1 1
3
2. (23, 24), y 5 22x 2 1
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continued
For use with pages 318–324
EXAMPLE 2
Determine whether lines are parallel or perpendicular
Determine which of the following lines, if any, are parallel or
2
perpendicular: Line a: 4y 2 6y 5 28, Line b: y 5 2}
x 1 1,
3
Line c: 2x 1 3y 5 215.
Solution
Find the slopes of the lines.
2
Line b: The equation is in slope-intercept form. The slope is 2}3 .
Write the equations for lines a and c in slope-intercept form.
LESSON 5.5
Line a: 4y 2 6x 5 28
Line c: 2x 1 3y 5 215
4y 5 6x 2 8
3y 5 22x 2 15
3
y 5 2}3 x 2 5
2
y 5 }2 x 2 2
3
2
Lines b and c have a slope of 2}3 , so they are parallel. Line a has a slope of }2, the
2
negative reciprocal of 2}3, so it is perpendicular to lines b and c.
Write an equation of a perpendicular line
Write an equation of the line that passes through (22, 1) and is
1
3
perpendicular to the line y 5 2} x 1 2.
Solution
STEP 1
1
Identify the slope. The graph of the given equation has a slope of 2}3.
Because the slopes of perpendicular lines are negative reciprocals, the
slope of the perpendicular line through (22, 1) is 3.
STEP 2
STEP 3
Find the y-intercept. Use the slope and the given point in y 5 mx 1 b.
1 5 3(22) 1 b
Substitute 3 for m, 22 for x, and 1 for y.
75b
Solve for b.
Write the equation. Use y 5 mx 1 b.
y 5 3x 1 7
Substitute 3 for m and 7 for b.
Exercises for Examples 2 and 3
3. Determine which of the following lines, if any, are parallel or perpendicular.
Line a: 23x 2 12y 5 36 Line b: x 1 4y 5 2 Line c: y 5 4x
4. Write an equation of the line that passes through (5, 3) and is perpendicular to
the line y 5 25x 1 3.
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Chapter 5 Resource Book
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EXAMPLE 3
Name ———————————————————————
LESSON
5.6
Date ————————————
Study Guide
For use with pages 325–333
GOAL
Make scatter plots and write equations to model data.
Vocabulary
A scatter plot is a graph used to determine whether there is a
relationship between paired data.
Correlation
If y tends to increase as x increases, the paired data are said to have a
positive correlation.
If y tends to decrease as x increases, the paired data are said to have a
negative correlation.
If x and y have no apparent relationship, the paired data are said to have
relatively no correlation.
When data show a positive or negative correlation, you can model the
trend in the data using a line of fit.
EXAMPLE 1
Describe the correlation of data
a.
y
40
35
30
25
20
15
0
b.
Hawk and Rabbit Population
Hawks sighted
(per square mile)
Rabbits (per square mile)
Rabbit Population and
Snow Depth
0 5 10 15 20 25 30 35 x
Snow depth (inches)
y
12
10
8
6
4
2
0
LESSON 5.6
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Describe the correlation of the data graphed in the scatter plot. The
data on the graph shows the results of a forest ranger’s annual survey
on rabbit and red-tail hawk populations.
0 10 20 30 40 50 60 70 x
Rabbits counted
(per square mile)
Solution
a. The scatter plot shows a negative
correlation between the depth of
snow cover during the winter and
the number of rabbits counted in
the spring. This means that as the
depth of the snow increased, the
number of rabbits counted
decreased.
b. The scatter plot shows a positive
correlation between the number of
hawks sighted and the number of
rabbits counted. This means that
as the number of rabbits increased,
the number of hawks increased.
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Study Guide
Date ————————————
continued
For use with pages 325–333
EXAMPLE 2
Make a scatter plot
Softball The table shows the number of girls signed up for a summer softball league
each year for 5 years.
Year
2001
2002
2003
2004
2005
Players
105
113
120
132
148
a. Make a scatter plot of the data.
b. Describe the correlation of the data.
c. Write an equation that models the number of girls signed up for
a summer softball league as a function of the number of years
since 2000.
Solution
y2 2 y1
148 2 100
Softball Sign-ups
Number of players
the number of years since 2000. Let y represent
the number of softball players. Plot the ordered
pairs as points in a coordinate plane.
b. The scatter plot shows a positive correlation,
which means that more players have signed up
each year since 2000.
c. Draw a line that appears to fit the points in the
scatter plot closely. Write an equation using
two points on the line. Use (1, 100) and (5, 148).
Find the slope of the line.
y
150
125
100
75
50
25
0
0 1 2 3 4 5 x
Years since 2000
48
m5}
5}
5}
5 12
x 2x
521
4
2
1
Find the intercept of the line. Use the point (5, 148).
y 5 mx 1 b
148 5 12(5) 1 b
88 5 b
Write slope-intercept form.
Substitute 12 for m, 5 for x, and 148 for y.
Solve for b.
An equation of the line of fit is y 5 12x 1 88.
Exercise for Example 1 and 2
1. a. Make a scatter plot of the data in the table.
x
1
2
2
3
3
4
4
5
y
9
7
6
4
3
3
2
1
b. Describe the correlation of the data.
c. Use the data in the table to write an equation that models y as
a function of x.
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Algebra 1
Chapter 5 Resource Book
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LESSON 5.6
a. Treat the data as ordered pairs. Let x represent
Name ———————————————————————
LESSON
5.7
Date ————————————
Study Guide
For use with pages 334– 342
GOAL
Make predictions using best-fitting lines.
Vocabulary
The line that most closely follows a trend in data is called the bestfitting line.
Using a line or its equation to approximate a value between two known
values is called linear interpolation.
Using a line or its equation to approximate a value outside the range of
known values is called linear extrapolation.
A zero of a function y 5 f (x) is an x-value for which f (x) 5 0
(or y 5 0).
EXAMPLE 1
Interpolate using an equation
Year
Pairs of Nesting Eagles
1998
2000
2001
2002
5
12
18
25
a. Make a scatter plot of the data.
b. Find an equation that models the number of pairs of nesting Bald Eagles as a
function of the number of years since 1998.
c. Approximate the number of nesting eagle pairs in 1999.
Solution
LESSON 5.7
a. Enter the data into lists on a graphing
calculator. Make a scatter plot, letting
the number of years since 1998 be the
x-values (0, 2, 3, 4) and the number of
nesting eagle pairs be the y-values.
b. Perform a linear regression using the
paired data. The equation of the bestfitting line is approximately
y 5 5x 1 4.
c. Graph the best-fitting line. Use the
trace feature and the arrow keys to
find the value of the equation
when x 5 1.
There were about 9 nesting pairs in 1999.
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Algebra 1
Chapter 5 Resource Book
X=1
Y=8.8563
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Nesting Eagles The table shows the number of pairs of nesting Bald Eagles at a
national wildlife reserve from 1998 to 2002.
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LESSON
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continued
For use with pages 334– 342
EXAMPLE 2
Extrapolate using an equation
Nesting Eagles Look back at Example 1.
a. Use the equation from Example 1 to approximate the number of nesting
Bald Eagle pairs in 2003 and 2005.
b. In 2003 there were actually 31 pairs of nesting eagles. In 2005 there were
actually 34 pairs of nesting eagles. Describe the accuracy of the extrapolations
made in part (a).
Solution
a. Evaluate the equation of the best-fitting line from Example 1 for x 5 5 and
x 5 7.
The model predicts about 29 pairs in 2003 and about 39 pairs for 2005.
b. The difference between the predicted number of pairs and the actual number of
pairs are 2 and 5, respectively. The difference in actual and predicted numbers
increased from 2003 to 2005. So, the equation of the best-fitting line gives a less
accurate prediction for years farther from the given data.
Exercise for Examples 1 and 2
1. Sales The table shows the sales (in thousands of dollars) of pet care items at
one pet store during the period 1998 to 2003.
Dollar amount
spent (thousands)
1998
2000
2002
2003
21
25
32
37
a. Find an equation that models the sales (in thousands of dollars) of pet
care items as a function of the number of years since 1998.
b. Predict the sales of pet care items for the years 2001 and 2004. Tell
whether the prediction is an interpolation or an extrapolation.
c. Which of the predictions from part (b) would you expect to be more
accurate? Explain your reasoning.
LESSON 5.7
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Year
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Chapter 5 Resource Book
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