u sing d irect and i nverse v ariation
DESCRIPTION
y. =. k ,. x. U SING D IRECT AND I NVERSE V ARIATION. D IRECT V ARIATION. The variables x and y vary directly if, for a constant k,. or y = kx,. k 0. k. =. y ,. x. U SING D IRECT AND I NVERSE V ARIATION. I NDIRECT V ARIATION. - PowerPoint PPT PresentationTRANSCRIPT
USING DIRECT AND INVERSE VARIATION
DIRECT VARIATION
The variables x and y vary directly if, for a constant k,
yx
= k,
k 0.
or y = kx,
USING DIRECT AND INVERSE VARIATION
INDIRECT VARIATION
The variables x and y vary inversely, if for a constant k,
kx
= y,
k 0.
or xy = k,
USING DIRECT AND INVERSE VARIATION
MODELS FOR DIRECT AND INVERSE VARIATION
DIRECT VARIATION INVERSE VARIATION
y = kx
k > 0
kx=y
k > 0
Using Direct and Inverse Variation
x and y vary directly
SOLUTION
yx
= k
42
= k
2 = k
When x is 2, y is 4. Find an equation that relates x and y in each case.
Write direct variation model.
Substitute 2 for x and 4 for y.
Simplify.
An equation that relates x and y is = 2, or y = 2x. yx
Using Direct and Inverse Variation
x and y vary inversely
SOLUTION
xy = k
(2)(4) = k
8 = k
When x is 2, y is 4. Find an equation that relates x and y in each case.
Write inverse variation model.
Substitute 2 for x and 4 for y.
Simplify.
An equation that relates x and y is xy = 8, or y = . 8x
Comparing Direct and Inverse Variation
SOLUTION
Inverse Variation: k > 0.As x doubles (from 1 to 2),y is halved (from 8 to 4).
8 xy =
x
y = 2x
1 2 3 4
2 4 6 8
8 4 283
Compare the direct variation model and the inverse variation model you just found using x = 1, 2, 3, and 4.
Make a table using y = 2x and y = .8 x
Direct Variation: k > 0.As x increases by 1,y increases by 2.
SOLUTION
Compare the direct variation model and the inverse variation model you just found using x = 1, 2, 3, and 4.
Comparing Direct and Inverse Variation
Plot the points and then connect the points with a smooth curve.
Inverse Variation: The graph for this model is a curve that gets closer and closer to the x-axis as x increases and closer and closer to the y-axis as x gets close to 0.
Direct Variation: the graph for this model is a line passing through the origin. Direct
y = 2x
Inverse8xy =
BICYCLING A bicyclist tips the bicycle when making turn. The angle B of the bicycle from the vertical direction is called the banking angle.
USING DIRECT AND INVERSE VARIATION IN REAL LIFE
Writing and Using a Model
banking angle, B
BICYCLING The graph below shows a model for the relationshipbetween the banking angle and the turning radius for a bicycle traveling at a particular speed. For the values shown, the banking angle B and the turning radius r vary inversely.
Writing and Using a Model
r
turning radius
banking angle, B
Turning Radius
Ban
kin
g an
gle
(deg
rees
)
Turning Radius
Ban
kin
g an
gle
(deg
rees
)
Writing and Using a Model
Find an inverse variation model that relates B and r.
Use the model to find the banking angle for a turning radius of 5 feet.
Use the graph to describe how the banking angle changes as the turning radius gets smaller.
r
turning radius
banking angle, B
Writing and Using a Model
From the graph, you can see that B = 32° when r = 3.5 feet.
B = kr
32 = k3.5
112 = k
SOLUTION
Turning Radius
Ban
kin
g an
gle
(deg
rees
)
Write direct variation model.
Substitute 32 for B and 3.5 for r.
Solve for k.
Find an inverse variation model that relates B and r.
The model is B = , where B is in degrees and r is in feet.112
r
Writing and Using a Model
SOLUTION
Use the model to find the banking angle for a turning radius of 5 feet.
Substitute 5 for r in the model you just found.
B = 1125
= 22.4
Turning Radius
Ban
kin
g an
gle
(deg
rees
)
When the turning radius is 5 feet, the banking angle is about 22°.
Writing and Using a Model
Use the graph to describe how the banking angle changes as the turning radius gets smaller.
SOLUTION
As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles.
Notice that the increase in the banking angle becomes more rapid when the turning radius is small.
As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles.As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles.As the turning radius gets smaller, the banking angle becomes greater. The bicyclist leans at greater angles.