Power Functions
Day 28: March 17, 2014
Notes
1. Which of the “bridges” below is likely to support the greatest weight, and which will support
the least?
2. Explain the reasoning behind your answer in #1.
3. If W represents the safe carrying weight; L, the length; and T, the thickness, what do you
expect to happen to the safe carrying weight if
a. The length increases while the value of the thickness remains constant?
b. The thickness increases while the value of the length remains constant?
4. Which of the following equations would best represent the type of relationship among the
three variables, and WHY?
a. W  L  T
c. W  L  T
d. W  L  T
T
b. W 
L
Definition: Directly Proportional Power Function
Definition: Indirectly Proportional Power Function
Definition: ‘k’
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Power Functions
Day 28: March 17, 2014
5. The area, A, of a circle as a function of the radius, r, is given by A  r    r .
2
a. Complete the sentence below.
 As the radius increases, the area will ___________________.
b. Identify the values of k and p.
c. Complete the sentence with either directly or indirectly:
 The area of a circle is ______________ proportional to the square of the radius.
6. An astronaut’s weight in pounds, W, as a function of her distance, d, in thousands of miles,
from the center of the earth is given by W  d  
2880
.
d2
a. Complete the sentence below.
 As the distance increases, the weight will ___________________.
b. Identify the values of k and p.
c. Complete the sentence with either directly or indirectly:
 The weight is ______________ proportional to the square of the distance.
7. The number of species, N, on an island as a function of the area, A, of the island is given by
N  A   4.13 A .
a. Complete the sentence below.
 As the area increases, the number of species will ___________________.
b. Identify the values of k and p.
c. Complete the sentence with either directly or indirectly:
 The number of species is ______________ proportional to the cube root of the area.
8. Express each of the following relationships with an equation. Be sure to include the
constant of proportionality, k.
a. The natural walking speed s of a person or animal is directly proportional to the square
root of their leg length L.
b. The heat, H, experienced from a hiker at a campfire is indirectly proportional to square of
the distance, d, between them.
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Power Functions
Day 28: March 17, 2014
c. The volume of a prism is jointly proportional to its length, width and height.
9. The circulation time of a mammal (that is, the average time it takes for all the blood in the
body to circulate once and return to the heart) is directly proportional to the fourth root of the
body mass of the mammal.
a. Write the general equation to describe the circulation time in seconds, T, as a function of
the body mass in pounds, M.
b. If an elephant of body mass 11,530 pounds has a circulation time of 148 seconds, find
the constant of proportionality and re-write the equation using this value. Round to two
decimal places.
c. What is the circulation time of a human with body mass 150 pounds?
10. The loudness of a sound measured in decibels (dB) is indirectly proportional to the square of
the distance from the source of the sound.
a. Write the general equation to describe the loudness, L, as a function of the distance, d.
b. A person who is 10 ft from a lawn mower experiences a sound level of 70dB. Using the
equation you wrote in part a, find the constant of proportionality and write the specific
equation that represents this situation.
c. Use the specific equation in part b to find the sound level when the person is 100 ft
away.
11. Circle the functions below that are power functions.
6
x4
a. y  3 x 5
b. y  3 x 5  1
c.
y
d. y  3  5 x
e. y  2x 4
f.
y 8 x
g. y  10 x  3
h. y  10 x
12. Which power functions from #11 are directly proportional and which functions are indirectly
proportional?
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Power Functions
Day 28: March 17, 2014
Visualizing Directly Proportional Power Functions (Part 1) – DUE NEXT CLASS!!
Recall that
x2  x  x
x3  x  x  x
1. Simplify the following:
a.
 2 
2

b.
 2 
3

2. Complete the statements below:
 When a negative number is raised to an even power, the answer will be ____________.

When a negative number is raised to an odd power, the answer will be _____________.
3. Fill in the table below and then graph the functions.
𝑥
f  x   x2
g  x   x3
−4
−3
−2
−1
0
1
2
3
4
f  x   x2
g  x   x3
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Power Functions
Day 28: March 17, 2014
y
y
25
25
20
20
15
15
10
10
5
5
x
-4
-3
-2
-1
1
2
3
x
4
-4
-3
-2
-1
1
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
2
3
4
4. Graph the following functions on your graphing calculator: y  x 2 , y  x 4 , and y  x 6 .
Sketch the shape of all even powered directly proportional functions.
5. Graph the following functions on your graphing calculator: y  x 3 , y  x 5 , and y  x 7 .
Sketch the shape of all odd powered directly proportional functions.
6. Fill in the table below and then graph the functions.
x
−4
−3
−2
−1
0
1
2
3
4
9
hx  x
Not possible –
you can’t take a
square root of a
negative number
hx  x
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Power Functions
Day 28: March 17, 2014
y
8
6
4
2
x
-9 -8 -7 -6 -5 -4 -3 -2 -1
1
2
3
4
5
6
7
8
9
-2
-4
-6
-8
1
x , y  x 4 , and y  x 0.7 .
Look at the shape only in Quadrant I. Sketch the shape of all directly proportional
functions with an exponent between 0 and 1 (in Quadrant I).
7. Graph the following function on your graphing calculator: y 
3
6