Direct Variation
Math II
Unit 7 Functions
Definition: direct variation
A direct variation is a function in the
form y = kx where k does not equal 0.
When we talk about a direct variation, we
are talking about a relationship where
as
x increases,
y increases
or as
x decreases,
y decreaeses at a CONSTANT RATE.
Direct Variation Model
The two variables x and y are said to
vary directly if their relationship is:
y = kx
k is the same as m (slope) k is called
the constant of variation
Is an Equation a Direct Variation?
If it is, find the constant of variation.
5x + 2y = 0
5
y  x
2
Solve for y.
1. Subtract 5x.
2. Divide by 2.
Yes, it’s a direct variation.
Constant of variable, k, is 5
 .
2
Is an Equation a Direct Variation?
If it is, find the constant of variation.
5x + 2y = 9
9  5x
y
2

5
y  4.5  x
2
Solve for y.
1. Subtract 5x.
2. Divide by 2.
No, it’s not a direct variation.
It’s not in the form y = kx.
The price of hot dogs varies directly with the
number of hotdogs you buy
You buy hotdogs.
x represents the number of hotdogs you buy.
y represents the price you pay.
y = kx
Let’s figure out k, the price per hotdog.
Suppose that when you buy 7 hotdogs, it costs $21.
Plug that information into the model to solve for k.
y = kx
21 = k(7)
7
k=3
Now divide both sides by 7
to solve for k.
7
The price per hotdog is $3.
y = 3x You could use this model to find the price (y) for any number of
hotdogs (x) you buy.
y
The graph of y = 3x
goes through the origin.
All direct variation graphs go
through the origin, because
when x = 0, y= 0 also.
x
y (price)
y = 3x
.
.
.
.
(3,9) When you buy 3 hotdogs, you pay $9
(2,6) When you buy 2 hotdogs, you pay $6
(1,3) When you buy 1 hotdog, you pay $3
x (number of hotdogs)
(0,0) When you buy 0 hotdogs, you pay $0
Finding the Constant of Variation (k)
STEPS
1. Plug in the known values for x and y
into the model: y = kx
2. Solve for k
3. Now write the model y = kx and
replace k with the number
4. Use the model to find y for other
values of x if needed
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1.
y  kx
84  k (24)
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1.
y  kx
84  k (24)
84 k (24)

24
24
7
k
2
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1.
y  kx
84  k (24)
84 k (24)

24
24
7
k
2
7
y x
2
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1.
y  kx
84  k (24)
84 k (24)

24
24
7
k
2
7
y x
2
2.
7
y x
2
Example
The variables x and y vary directly. When x = 24, y = 84.
1. Write the direct variation model that relates x and y.
2. Find y when x is 10.
1.
y  kx
84  k (24)
84 k (24)

24
24
7
k
2
7
y x
2
2.
7
y x
2
7
y  10 
2
y  35
When x = 10, y = 35
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
1
18  k  
2
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
1
18  k  
2
1
(2)18  k   (2)
2
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
1
18  k  
2
1
(2)18  k   (2)
2
36  k
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
1
18  k  
2
1
(2)18  k   (2)
2
36  k
y  36 x
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
1
18  k  
2
1
(2)18  k   (2)
2
36  k
y  36 x
2.
y  36 x
Example
The variables x and y vary directly. When x = ½, y = 18.
1. Write the direct variation model that relates x and y.
2. Find y when x is 5.
1.
y  kx
1
18  k  
2
1
(2)18  k   (2)
2
36  k
y  36 x
2.
y  36 x
y  36 (5)
y  180
When x = 5, y = 180
Real-World Example
Your distance from lightning varies directly
with the time it takes you to hear thunder. If
you hear thunder 10 seconds after you see
the lightning, you are about 2 miles from the
lightning. Write an equation for the
relationship between time and distance.
Real-World Example Continued….
Relate:
The distance varies directly with the time. When
x = 10, y = 2.
Define:
Let x = number of seconds between seeing
lightning and hearing thunder.
Let y = distance in miles from lightning.
Real-World Example Continued…
y = kx
2 = k(10)
1
k
5
Use general form of direct variation.
Substitute 2 for y and 10 for x.
Solve for k.
Write an equation using the value for k.
1
y x
5
Example
A recipe for a dozen corn muffins calls for 1
cup of flower. The number of muffins varies
directly with the amount of flour you use.
Write a direct variation for the relationship
between the number of cups of flour and
the number of muffins.
Example
The force you must apply to lift an object
varies directly with the object’s weight. You
would need to apply 0.625 lb of force to a
windlass to lift a 28-lb weight. How much
force would you need to lift 100 lb?
Example:
A car uses 8 gallons of gasoline to travel 290
miles. How much gasoline will the car use to
travel 400 miles?
Example:
Julio wages vary directly as the number of hours
that he works. If his wages for 5 hours are
$29.75, how much will they be for 30 hours?
Inverse Variation
Math II
Unit 7 Functions
Inverse Variation
Inverse is very similar to direct, but in
an inverse relationship as
one value goes up, the other goes down.
There is not necessarily a constant
rate.
Definition: inverse variation
An inverse variation is a function in the
form y = k
x
where k does not equal 0
where k is still the constant of variation.
Graphing Inverse Variations
An inverse variation can also be
identified by its graph. Some
inverse variation graphs are
shown. Notice that each graph
has two parts that are not
connected.
Also notice that none of the
graphs contain (0, 0). In other
words, (0, 0) can never be a
solution of an inverse variation
equation.
Example
The variables x and y vary inversely. When x = -12, y = 0.5.
1. Write the inverse variation model that relates x and y.
2. Find y when x is 3.
Example
If y varies inversely as x and x = 18 when y
= 6, find y when x = 8.
Example
If y varies inversely with x and
y = 12 when x = 2, find y when x = 8.
Example
Y varies inversely with x. If y = 40 when
x = 16, find x when y = -5.
Example
The cost per person to rent a mountain cabin is
inversely proporitonal (varies inversely) to the
number of people who share the rent. If the cost is
$36 per person when 5 people share, what is the
cost per person when 8 people share?
Example
The time it takes to fly from Los Angeles to New
York varies inversely as the speed of the place. If
the trip takes 6 hours at 900 km/h, how long would
it take at 800 km/h?
Example
The volume V of gas varies inversely to the
pressure P. The volume of a gas is 200 cm3 under
pressure of 32 kg/cm2. What will be its volume
under pressure of 40 kg/cm2?
Joint Variation
Math II
Unit 7 Functions
Inverse Variation
There are situations when more than one
variable is involved in a direct variation
problem.
In these cases the problem is called joint
variation.
The equation remains the same except that
additional variables are included in the
product.
For example if z varies jointly with the values
of x and y the equation will be
z = kxy
.
Example:
Variable I varies jointly as the values of P
and T.
If I = 1200 when P = 5000 and T = 3 ,
find I when P = 7500 and T = 4.
Example:
The cost c of materials for a deck varies jointly
with the width w and the length l.
If c = $470.40 w hen w = 12 and l = 16 , find
the cost when w = 10 and l = 25.
Example:
The value of real estate V varies jointly with
the neighborhood index N and the square
footage of the house S.
If V = $376, 320 when N = 96 and S = 1600 ,
find the value of a property with N = 83 and
S = 2150.
Example:
The number of gallons g in a circular swimming
pool varies jointly with the square of the radius
r2 and the depth d.
If g = 754 when r = 4 and d = 2 , find the
number of gallons in the pool when r = 3 and d =
1.5.
EXAMPLE*
Joint – Direct AND Inverse!
To build a sound wall along the highway, the amount of
time t needed varies directly with the number of
cement blocks c needed and inversely with the
number of workers w.
A sound wall made of 2400 blocks, using six workers
takes 18 hours to complete. How long would it take to
build a wall of 4500 blocks with 10 workers?
EXAMPLE*
Joint – Direct AND Inverse!
The number of hours needed to assemble computers
varies directly as the number of computers and
inversely as the number of workers. If 4 workers can
assemble 12 computers in 9 hours, how many workers
are needed to assemble 48 computers in 8 hours?
EXAMPLE*
An egg is dropped from the roof of a building. The
distance it falls varies directly with the square of the
time it falls. If it takes 4 seconds for the egg to fall
15 feet, how long will it take the egg to fall 500 feet?