Download BENCHMARK 2 E. Slope-Intercept Form and Direct

Name ———————————————————————
Date ————————————
BENCHMARK 2
(Chapters 3 and 4)
E. Slope-Intercept Form and Direct
Variation (pp. 32–35)
For any two points, there is one and only one line that contains both points. This fact
can help you graph a linear equation. Many times, it will be convenient to use the
points where the line crosses the x-axis and y-axis. These points are the intercepts.
Knowing how steep the line is, or the slope of the line, also can help you graph a linear
equation. If the graph of a linear equation passes through the origin (0, 0), the relationship between x and y is called a direct variation.
Vocabulary
EXAMPLE
x-intercept The x-coordinate of the point where a graph intersects the x-axis.
y-intercept The y-coordinate of the point where a graph intersects the y-axis.
Find the x-intercept and the y-intercept of the graph of 3x 4y 12.
Solution:
To find the x-intercept, substitute 0 for y and solve for x.
3x 1 4y 5 12
Write original equation.
3x 1 4(0) 5 12
12
x5}
54
3
Remember
that the x- and
y-intercepts are
numbers, NOT
ordered pairs.
Substitute 0 for y.
Solve for x.
To find the y-intercept, substitute 0 for x and solve for y.
3x 1 4y 5 12
Write original equation.
3(0) 1 4y 5 12
12
y5}
53
4
Substitute 0 for x.
Solve for y.
The x-intercept is 4. The y-intercept is 3.
PRACTICE
Find the x-intercept and the y-intercept of the graph of the equation.
1.
x 1 y 5 26
4. 27y 5 14x
2.
23y 1 8 5 212x
3. 4.5x 1 0.5y 5 9
5.
215 1 10y 5 60x
6. 3 2 18x 5 26y
2. Find the Slope of a Line
Vocabulary
Slope Describes how quickly a line rises or falls as it moves from left to right. Slope
is the ratio m of the vertical change between two points on the line to the horizontal
change between the same two points.
y2 2 y1
For points (x1, y1) and (x2, y2), m 5 }
x 2x .
2
EXAMPLE
32
1
Find the slope of the line that passes through the points.
a. (1, 5) and (4, 6)
b. (25, 7) and (3, 21)
c. (22, 7) and (8, 7)
d. (6, 28) and (6, 2)
Algebra 1
Benchmark 2 Chapters 3 and 4
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
BENCHMARK 2
E. Slope-Intercept Form
1. Find the Intercepts of the Graph of an Equation
Name ———————————————————————
Date ————————————
BENCHMARK 2
(Chapters 3 and 4)
Solution:
Think of (x1, y1) as
“the coordinates
of the first point”
and (x2, y2) as “the
coordinates of the
second point.” Be
sure to subtract
the x- and
y-coordinates in
the same order.
a. Let (x1, y1) 5 (1, 5) and (x2, y2) 5 (4, 6).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
625
1
5}
5 }3
421
b.
Substitute and simplify.
Let (x1, y1) = (25, 7) and (x2, y2) 5 (3, 21).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
28
Substitute and simplify.
BENCHMARK 2
E. Slope-Intercept Form
21 2 7
3 2 (25)
5}5}
5 21
8
c. Let (x1, y1) 5 (22, 7) and (x2, y2) 5 (8, 7).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
727
8 2 (22)
0
5}5}
50
10
Substitute and simplify.
The slope is 0. The line is horizontal.
d. Let (x1, y1) 5 (6, 28) and (x2, y2) 5 (6, 2).
y2 2 y 1
m5}
x 2x
2
Write formula for slope.
1
2 2 (28)
10
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5}
5}
626
0
Substitute. Division by 0 is undefined.
The slope is undefined. The line is vertical.
PRACTICE
Find the slope of the line that passes through the points.
7. (6, 29) and (29, 6)
9. (211, 8) and (13, 5)
11. (2.5, 25) and (5.5, 29)
8. (4, 2) and (4, 0)
10. (21, 27) and (1, 27)
12. (23, 25) and (22, 0)
3. Graph an Equation Using Slope-Intercept Form
Vocabulary
EXAMPLE
Slope-intercept form A linear equation in the form y 5 mx 1 b, where m is the slope
and b is the y-intercept of the graph of the equation.
Graph the equation x 2y 4.
Solution:
If you can
substitute the
coordinates of
the second point
in the original
equation and get
a true statement,
then your graph is
correct.
Step 1: Rewrite the equation in slope-intercept form.
1
y 5 }2 x 1 2
Step 2: Identify the slope and the y-intercept.
1
m 5 }2 and b 5 2.
Step 2: Plot the point that corresponds to the
y-intercept, (0, 2).
Step 4: Use the slope to find another point on the line.
Draw a line through the two points.
6
5
4
y
3
2
(0, 2)
1
24 23 22 21
21
1 2
(2, 3)
3
4 x
22
Algebra 1
Benchmark 2 Chapters 3 and 4
33
Name ———————————————————————
Date ————————————
BENCHMARK 2
(Chapters 3 and 4)
PRACTICE
Graph the equation.
2
13. y 5 2} x 1 7
5
16. y 5 24
14.
23x 5 4y 1 8
15. 3x 2 3y 5 6
17.
214x 2 7y 5 21
18. 1.5y 2 6x 2 12 5 0
4. Identify Direct Variation Equations
BENCHMARK 2
E. Slope-Intercept Form
Vocabulary
EXAMPLE
Direct variation An equation in the form y 5 ax, where a Þ 0, represents direct
variation. The variable y varies directly with x.
Constant of variation The constant a in the direct variation equation y 5 ax.
Tell whether the equation represents direct variation. If so, identify the
constant of variation.
a. 6x 2 4y 5 0
b. x 1 y 5 8
Solution:
Try to rewrite the equation in the form y 5 ax.
a. 6x 2 4y 5 0
Write original equation.
24y 5 26x
3
y 5 }2 x
Subtract 6x from each side.
Simplify.
Because the equation 6x 2 4y 5 0 can be rewritten in the form y 5 ax,
3
b. x 1 y 5 8
Write original equation.
y 5 2x 1 8
Subtract x from each side.
Because the equation x 1 y 5 8 cannot be rewritten in the form y 5 ax, it does
not represent direct variation.
PRACTICE
Tell whether the equation represents direct variation. If so, identify the
constant of variation.
7
19. y 5 2} x
20. x 1 4 5 16y
21. 9y 5 5x
8
22. x 5 247y
23.
23 1 x 1 7 5 2y 1 4 24. 13 5 26x
5. Write and Use a Direct Variation Equation
EXAMPLE
The graph of a direct variation equation is shown.
a. Write the direct variation equation.
b. Find the value of y when x 5 36.
5
4
23
Algebra 1
Benchmark 2 Chapters 3 and 4
(6, 5)
3
2
1
22 21
21
22
34
y
1 2
3
4
5
6 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
it represents direct variation. The constant of variation is }2 .
Name ———————————————————————
Date ————————————
BENCHMARK 2
(Chapters 3 and 4)
PRACTICE
Solution:
a. Because y varies directly x, the equation has the form y 5 ax. Use the fact that
y 5 5 when x 5 6 to find a.
y 5 ax
Write direct variation equation.
5 5 a(6)
Substitute.
5
}5a
6
Solve for a.
5
A direct variation equation that relates x and y is y 5 }6 x.
5
b. When x 5 36, y 5 } (36) 5 30.
6
Write the direct variation equation that passes through the given point.
Then find the value of y for the given x.
25. (3, 21); x 5 12
26.
(24, 28); x 5 32
27. (26, 3); x 5 18
28. (9, 2); x 5 27
29.
(25, 7); x 5 100
30. (22, 21); x 5 74
Quiz
BENCHMARK 2
E. Slope-Intercept Form
Check the sign of
the constant of
variation in your
equation. If the
graph of y 5 ax
passes through
Quadrants I and
III, the constant
should be positive.
If the graph of
y 5 ax passes
through Quadrants
II and IV, the
constant should
be negative.
Find the x-intercept and the y-intercept of the graph of the equation.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1. 221 1 14y 5 84x
2.
23 1 x 5 3y
3. 3.2x 1 0.8y 5 4
Find the slope of the line that passes through the points.
4. (8, 25) and (23, 4)
5.
(1, 7) and (22, 7)
6. (29, 7) and (3, 25)
8.
y 5 22
9. 4x 2 6y 5 12
Graph the equation.
7. y 5 x 1 1
Does the equation represent direct variation? If so, find the constant of
variation.
4
10. y 5 2} x
11. x 1 3 5 9y
12. 4y 5 7x
5
Write the direct variation equation that passes through the given point.
Then find the value of y for the given x.
13. (2, 25); x 5 20
14.
(23, 29); x 5 43
15. (24, 6); x 5 64
Algebra 1
Benchmark 2 Chapters 3 and 4
35