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4-2 Warm Up. Determine the slope of the line that passes through each pair of points: (3, 5) and (7, 12) (-2, 4) and (5, 4) (-3, 6) and (2, -6) (7, -2) and (7, 13). - PowerPoint PPT Presentation
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Determine the slope of the line that passes through each pair of points:
(3, 5) and (7, 12) (-2, 4) and (5, 4)
(-3, 6) and (2, -6) (7, -2) and (7, 13)
Determine the value of n so that the slope of the
line through (n, 4) and (1, n) is . 21
47
m 0m
512
m undefined
3n
Algebra 1 Glencoe McGraw-Hill JoAnn Evans
Math 8H
4-2
Slope and
Direct Variation
A direct variation equation is a special type of linear equation.
Every direct variation equation will graph as a line that passes through the origin. (0,
0)
x
y
The two quantities will be represented as y and x.
Written in ratio form the ratio of y to x is .
What ratio did we study in the previous lesson? SLOPE!
In direct variation equations, the slope has a different name. It is known as the “constant of
variation”.
When two quantities have a constant ratio, they are said to have a direct variation.
xy
Slope is the ratio of the change in y to the change in
x.change in y
m (SLOPE)change in x
A direct variation equation is: kxy
In a direct variation equation k is called the constant of variation. On
the graph of a direct variation equation k is the slope of the line.
Solve the direct variation equation for y.
kxy
kxy
)x()x(
kxy
A direct variation equation represents a constant rate of change.
“k” is the constant of variation
This is a graph of the direct variation
equation y = 3x.
The constant of variation is 3.
What is the slope of the line?
12
12
xxyy
313
0103
(1, 3)
(0, 0)
The slope of the line is the same
as the constant of variation.
This is a graph of the
direct variation equation
y = x.
What is the constant of
variation?
What is the slope of the line?
12
12
xxyy
41
0401
(4, 1)
(0, 0)
The slope of the line is the same
as the constant of variation.
41
41
Remember: every direct variation equation will graph as a line that passes through
the origin.
x
y
Graph y = 5x
x
y
•
• 1. Write the slope as a ratio.
15
5
2. Plot a point at (0, 0).
3. Walk the slope. A
slope
of tells you to go
UP 5, OVER 1.
15
4. Plot the point. Connect the two
points with a line.
Graph y = x
1. The slope is already a ratio. Assign the negative to the numerator.2. Plot a point at (0, 0).
3. Walk the slope. A
slope
of tells you to go
DOWN 3, OVER 4.
43
4. Plot the point. Connect the two
points with a line.
x
y
••
43
Graph y = x
x
y
••
52
Graph y = -x
x
y
••
What is the slope?
It’s -1. Written as a
ratio, that’s . 11
Y varies directly as x. Write a direct variation equation that relates x and y.
kxy
Use this information towrite the direct variation
equation.
( 27) k( 3) 3 3
If y = -27 when x = -3, find x when y = 108.
Using the equation, answer the question.
y 9x108 9x
9 9
12 x
x equals 12 when y = 108.
k9
kxy x9y
Y varies directly as x. Write a direct variation equation that relates x and y.
kxy
Use this information towrite the direct variation
equation.
( 15) k(5) 5 5 k3
kxy x3y
If y = -15 when x = 5, find x when y = -87.
Using the equation, answer the question.
y 3x87 3x
3 3 x29
x equals 29 when y = -87.
Y varies directly as x. Write a direct variation equation that relates x and y.
kxy
Use this information towrite the direct variation
equation.
(7.5) k(.5) .5 .5
15 k
kxy y 15x
If y = 7.5 when x = 0.5, find y when x = -0.3.
Using the equation, answer the question.
y 15x
y 15 .3 y 4.5
y equals -4.5 when x = -.3.
Y varies directly as x. Write a direct variation equation that relates x and y.
kxy
Use this information towrite the direct variation
equation.
(12) k(18) 18 18
2k
3
kxy 2
y x3
If y = 12 when x = 18, find x when y = -16.
Using the equation, answer the question.
2y x
3
216 x
3
24 x
x equals 24when y = -16.
The cost of bananas varies directly with their weight. If 3 pounds of bananas cost $2.04, find the cost of 4
pounds.
c kw
2.04 k 3 3 3
.68 k
c .68w
If c = $2.04 when w = 3, find c when w = 4.
c .68w
c .68 4
c 2.72
The cost is $2.72 for 4 lb. of bananas.
Write a direct variation equation that relates the cost, c, to the weight, w. Use the equation to answer the question.
d = rt is a direct variation equation! Distance (d) varies directly as time (t).The rate (r) is the constant of variation.
A hot air balloon’s distance of ascent varies directly as the time. The balloon ascended 372
feet in six minutes. Write a direct variation equation that relates the distance, d, to the time,
t.d = rt
(372) = r(6)
62 = r The balloon’s ascent rate is 62 feet per minute.d = 62t is the direct variation
equation.
Use the direct variation equation to find how long will it take for the balloon to rise 1209 feet.
d = 62t (1209) = 62t
19.5 = t
The balloon should ascend 1209 feet in 19.5 minutes.