The Power Functions
Direct Variation
What is it and how do I know when I see it?
A power function - is any function in the
form of y = kxn, where k is nonzero and n is a
positive number (1, 2, and 3).
The linear equation graph
at the right shows that as
the x-value increases, so
does the y-value increase.
For instance, if x = 2, y = 4.
If x = 6 (multiplied by 3), then
y = 12 (also multiplied by 3).
Direct variation – y varies directly as x means
that y = kx, where k is the constant of
variation.
(see any similarities to y = mx + b ?)
• Another way of writing this is k = y/x
In other words:
• As x increases in value, y increase or
• As x decreases in value, y decrease.
This is a graph of direct
variation. If the value of
x is increased, then y
increases as well. Both
variables change in the
same manner.
If x decreases, so does
the value of y. It is said
that y varies directly as
the value of x.
Direct variation with a power
where n = 1.
Others are y =kx2 and y = kx3
A direct variation
between 2 variables, y and
x, is a relationship that is
expressed as:
y = kx
The relationship between y and
x that is expressed as
y = kx, k is called the constant
of proportionality.
In most problems, the k value
needs to be found using the
first set of data given.
Example:
The power, P, of a gear varies
directly with the radius, r, of a
gear. Find the constant of
proportionality if P = 300 when
r = 50
Start with the formula
P = kr
Typical problem, try it
In a factory, the profit, P, varies
directly with the inventory, I. If P =
100 when I = 20, find P when I = 50.
Step 1
Set up the formula
It will be necessary to
use the “first” set of
data to find the value
for the constant, k.
P = Ik
y = kx
Step 2
Find the missing constant, k, for the
given data.
100 = (20)k, k = 5
Step 3
Use the formula and constant to find
the missing value.
P = (50)(5) P = 250
x
y
6
12
7
8
14
16
Note: x increases
6, 7, 8
And y increases
12, 14, 16
What is the constant of variation of the table
above?
Since y = kx, we can say k = y/x therefore,
12/6 = k or k = 2 14/7 = k or k = 2
16/8 = k or k = 2 Note: k stays constant
y = 2x is the equation!!!
Another example
X
Y
30
10
15
9
5
3
Note: x decreases,
30, 15, 9
And y decreases.
10, 5, 3
What is the constant of variation of the table
above?
Since y = kx, we can say k = y/x therefore:
30/10 = k or k = 3 15/5 = k or k = 3
9/3 = k or k = 3
Note: k stays constant.
y = 3x is the equation
Another example
x
y
-4
-1
-16
-40
-4
-10
Note: x decreases,
-4, -16, -40
And y decreases.
-1, -4, -10
What is the constant of variation of the table
above?
Since y = kx, we can say k = y/x therefore:
-1/-4 = k or k = ¼ -4/-16 = k or k = ¼
-10/-40 = k or k = ¼
Note: k stays constant
y = ¼x is the equation
Use direct variation to solve word problems
• A car uses 8
gallons of gasoline
to travel 290 miles.
How much gasoline
will the car use to
travel 400 miles?
Step 2: find the
constant variation and
equation.
k = y/x or k = 290/8
or 36.25
y = 36.25x
Step 1: find points in the table
x
y
8
?
290
400
Step 3: use the equation
to find the unknown.
400 = 36.25x
400 = 36.25x
36.25 36.25
or x = 11.03
Direct variation and its graph
y = mx + b,
m = slope and
b = y-intercept
With direct variation the
equation is y = kx
Note: m = k or the constant
and b = 0, therefore the
graph will always go through…
The ORIGIN!!!!!
Solve the following variation problems using
the formula.
1) y varies directly as x. If y = 75 when x = 10, find y
when x = 16.
2) Your distance, d, from lightning varies directly with
the time, t, it takes you to hear thunder. If you
hear thunder 10 seconds after you see lightning,
you are about 2 miles from the lightning. How far
are you away from the lightning if you hear thunder
in 3 seconds.
3) The distance, d, a cyclist travels varies directly
with the time, t, it takes in hours. If a cyclist
travels 40 km in 2 hours, how far will he have
traveled in 6 hours.
More problems
4) The amount of sales tax on a new car is
directly proportional to the purchase price of
the car. If a $25,000 car cost $1750 in sales
tax, what is the purchase price of a new car
which has a $3500 sales tax?
Hint: sales tax = k(purchase price)
5) The cost of a house in Florida is directly
proportional to the size of the house. If a
2850 ft2 house cost $182,400, then what is
the cost of a 3640 ft2 house?
Other Power Functions
y = kx2 and y = kx3
y = kx2
The distance required for a moving car to stop varies
directly as the square of the car’s speed. Therefore,
the formula (equation) d = ks2 represents the stopping
distance for the auto. A car traveling 50 miles per
hour has a stopping distance of 135 feet. What would
be the stopping distance of an auto travel 70 miles per
hour.
d = ks2
d = ks2
135 = k(502)
d = .054(702)
k = .054
d = 264.6 ft
y = kx3
The volume of a cube varies directly as the cube of
the side lengths. If the volume (V) of a cube is
36cm with side lengths (s) of 2cm, what is the
volume of a cube with side lengths (s) of 5.
V = ks3
V = 4.5(53)
36 = k(23)
y = 562.5 cm
k = 4.5