Deck 7: Power Functions

Question 1

The function,
,
gives the perimeter of a square as a function of the length of a side.
By what factor does the perimeter increase when the length of a side is tripled?

Answer 1

3

Question 2

The function,
,
gives the area of a circle as a function of its radius.
By what factor does the area increase when its radius is quadrupled?

Answer 2

16

Question 3

The function,
,
gives the volume of a sphere as a function of its radius.
By what factor does the volume increase when its radius is quadrupled?

Answer 3

64

Question 4

The function,
,
Gives the area of a 10-inch long rectangle as a function of its width.
By what factor does the area increase when its width is quadrupled?

A) 40
B) 4
C) 0.4
D) 16

Question 5

The function,
,
Gives the area of a circle as a function of its radius.
By what factor does the area increase when its radius is doubled?

A) 2
B) 12
C) 4
D) 0.75

Question 6

The function,
,
Gives the volume of a cube as a function of the length of a side.
By what factor does the volume increase when the length of a side is quadrupled?

A) 85
B) 4
C) 3
D) 64

Question 7

In which function does the output, y, increase by a factor of 8 when the input, x, is doubled?

A)
B)
C)
D)

Question 8

In which function does the output, y, increase by a factor of 9 when the input, x, is tripled?

A)
B)
C)
D)

Question 9

In which function does the output, y, increase by a factor of 16 when the input, x, is quadrupled?

A)
B)
C)
D)

Question 10

Solve the formula, F = 2·m·a for m.

Question 11

Find the surface area of a sphere with a radius of 4 meters.
Round the answer to 2 decimal places.

Question 12

Find the Volume area of a sphere with a radius of 7 meters.
Round the answer to 2 decimal places.

Question 13

Find the radius of a sphere with a surface area of 113.10 feet².
Round the answer to the nearest foot.

Question 14

Find the radius of a sphere with a volume of 3,053.63 meters³.
Round the answer to the nearest meter.

Question 15

Find the volume of a cylinder with a radius of 6 ft and a height of 11 ft.
Round the answer to 2 decimal places.

Question 16

Find the surface area of a cylinder with a radius of 5 m and a height of 9 m.
Round the answer to 2 decimal places.

Question 17

A sphere with radius 12 m and a cylinder with radius 12 m are to have the same volume. What must be the height of the cylinder?

Question 18

Assume a box has a square base with a side of length x and the side of the base is 4 times the height. Write a function for the volume of the box.

Question 19

Assume a box has a square base, a height of x, and the side of the base is 4 times the height. Write a function for the volume of the box.

Question 20

A sculptor has been inspired by a teacher in his youth and now wants to commemorate the teacher by creating a sculptor of her. The sculptor first makes a model that is 12 inches tall and contains 16.8 cubic inches of clay. How many cubic inches of clay will be required for the final sculpture if it is to be 60 inches tall?

Question 21

Determine which one of the following functions is a power function.

A)
B)
C)
D)

Question 22

For the power function below identify A) k and B) p.

Question 23

Given that y is directly proportional to x and when , find a formula to describe the relationship.

Question 24

Given that y is directly proportional to x, and when , find y when .

Question 25

Given that y is directly proportional to x, and when ,

A) find a general formula to describe the relationship, and
B) use the formula to find y when .

Question 26

Given that y is directly proportional to the cube root of x, and when , find a formula to describe the relationship.
Write your answer in the form .

Question 27

Given that y is directly proportional to the square of x, and when , the formula describes this relationship.
Find A) p and B) k.

Question 28

Given that y is directly proportional to the square root of x, and when , the formula describes this relationship. Find p and k.

A) p is 19.5 and k is 15.21
B) p is 5 and k is 1/2
C) p is 15.21 and k is 19.5
D) p is 1/2 and k is 5

Question 29

Given that y is directly proportional to the cube of x, and when , find y when .
Round to two decimal places, if necessary.

Question 30

Given that y is directly proportional to the square of x, and when ,

A) find a formula to describe the relationship (Write your answer in the form ) and,
B) use the formula to find y when . Round your answer to 2 decimal places, if necessary.

Question 31

The cost, C, of a diamond is directly proportional to the square of its weight, w (in carats).
A 0.5-carat diamond costs 2,250 dollars.

A) Find a general formula to describe the relationship, (Write your answer in the form ), and
B) use the formula to find the cost, in dollars, of a 3.5-carat diamond.

Question 32

The cost, C, of a diamond is directly proportional to the square of its weight, w (in carats).
If a 1-carat diamond costs 8,500 dollars, find the cost, in dollars, of a 4-carat diamond.

Question 33

If y is directly proportional to the cube of x, by what factor does y change when x triples?

A)
B) 3
C)
D) 1/3

Question 34

If w is jointly proportional to the square root of x and to the square of y, and when and , then find a general formula to describe the relationship. Use fractional exponents rather than radicals, if necessary.

Question 35

If w is jointly proportional to the square of x and to y, and when and , then find w when and . Round your answer to 2 decimal places, if necessary.

Question 36

If w is jointly proportional to the cube of x and to the cube of y, and when and , then

A) find a general formula to describe the relationship (use fractional exponents rather than radicals, if necessary) and,
B) use the formula to find w when and . Round your answer to 2 decimal places, if necessary.

Question 37

The cost, C, of copper tubing is jointly proportional to the length, l, of the pipe and the square of the diameter, d, of the pipe.
A 32-inch length of 1-inch diameter copper tubing costs $48.00.

A) Find a general formula to describe the relationship, and
B) use the formula to find the cost, in dollars, of a 44-inch length of 1.5-inch diameter copper pipe.Round your answer to 2 decimal places, if necessary.

Question 38

Evaluate the following function when x = -2.

Question 39

Which of the following is the graph of a power function?

A)

B)

C)

Question 40

What description best fits the power function, , pictured?

A) k is negative and p is odd
B) k is positive and p is odd
C) k is negative and p is even
D) k is positive and p is even

Question 41

Which of the following is the graph of a power function, , where k is positive and p is odd?

A)

B)

C)

D)

Question 42

Find the description that best fits a power function, , where and .

A) k is negative and p is even
B) k is negative and p is odd
C) k is positive and p is even
D) k is positive and p is odd

Question 43

Find the description that best fits a power function, , where k is positive and p is odd.

A) and
B) and
C) and
D) and

Question 44

Which graph is the graph of a power function, , that has an odd p?

A) A
B) B
C) neither
D) both

Question 45

Which of the following describes the function, ?

A)
B)
C) both A and B
D) neither A or B

Question 46

Both graphs below are power functions of the form . Which graph has a larger k?

A) A
B) B

Question 47

Both graphs below are power functions of the form . Which graph has a larger k?

A) A
B) B

Question 48

Graph A is the graph of , and graph B is the graph of . What is the value of k?

Question 49

Graph A is the graph of , and graph B is the graph of . What is the value of k?

Question 50

Are f(x) = 3x⁹ and g(x) = -3x⁹ reflections of each other across the horizontal axis?

Question 51

Sketch a graph of the power function y = 3x³_.

Question 52

Sketch a graph of the power function y = 4x⁸_.

Question 53

The graph of a function of the form f(x) = -5x^p, is symmetrical about the horizontal axis when p is even.

Question 54

For and As which of the following is true?

A)
B)

Question 55

As which function dominates, or ?

A)
B)

Question 56

For 0 < x < 5, where do the graphs of f(x) = 9x⁹ and g(x) = 9^x intersect?

Question 57

Compare the functions f(x) = 2x² and g(x) = 2^x. On the interval (1, 5),

A) f(x) < g(x)
B) f(x) = g(x)
C) f(x) > g(x)

Question 58

Compare the functions f(x) = 6x⁶ and g(x) = 6^x. On the interval ,

A) f(x) < g(x)
B) f(x) = g(x)
C) f(x) > g(x)

Question 59

Which graph represents the function f(x) = 8x⁸?

A)

B)

C)

D)

Question 60

Which graph represents the function f(x) = 7x⁷?

A)

B)

C)

D)

Question 61

Which graph represents the function f(x) = -12x¹²?

A)

B)

C)

D)

Question 62

Which graph represents the function f(x) = 3x³?

A)

B)

C)

D)

Question 63

Which graph represents the function f(x) = 7^x?

A)
B)

C)

D)

Question 64

Which graph represents the function f(x) = 5^-x?

A)

B)

C)

D)

Question 65

Given the points (2, 24) and (4, 192), find the power function of the form y = ax^p that goes through the two points.

Question 66

Given the points (2, 4) and (4, 16), find the power function of the form y = a^x that goes through the two points.

Question 67

Given that y is inversely proportional to x, and y = 4 when x = 21. Find a general formula to describe the relationship.

Question 68

Given that y is inversely proportional to x, and y = 10 when x = 15. Find y when x is 25.

Question 69

Given that y is inversely proportional to the square of x and when , find a general formula to describe the relationship.
Write your answer with positive exponents.

Question 70

Given that y is inversely proportional to the cube root of x and when , the formula describes this relationship.
Find A) p and B) k.

Question 71

Given that y is inversely proportional to the square of x and when , the formula describes this relationship. Find p and k.

A) p is 54 and k is 2
B) p is 6 and k is 3
C) p is 2 and k is 54
D) p is 3 and k is 6

Question 72

Given that y is inversely proportional to the square root of x and when , find y when .
Round your answer to 2 decimal places, if necessary.

Question 73

Given that y is inversely proportional to the cube root of x and when ,

A) find a general formula to describe the relationship (write your answer with positive exponents) and,
B) use the formula to find y when . Round your answer to 2 decimal places, if necessary.

Question 74

The percentage of sunlight, s, that filters through water is inversely proportional to the depth of the water, d. If 10 meters below the surface only 25% of sunlight is visible, find what percent of sunlight will be visible 30 meters below the surface.
Round your answer to 2 decimal places, if necessary.

Question 75

Given that w is directly proportional to x and inversely proportional to y, and when and , find a formula to describe this relationship. Use only positive exponents and use fractional exponents rather than radicals, if necessary. Round values to 2 decimal places, if necessary.

Question 76

Given that w is directly proportional to the square root of x and inversely proportional to y, and when and , find w when and . Round your answer to 2 decimal places, if necessary.

Question 77

Given that w is directly proportional to the cube of x and inversely proportional to the square root of y and, when and ,

A) find a formula to describe the relationship (use only positive exponents and fractional exponents rather than radicals, if necessary) and,
B) use the formula to find w when and . Round your answer to 2 decimal places, if necessary.

Question 78

The amount of time needed to paint the interior of an office building is directly proportional to the square footage of the building and inversely proportional to the number of painters on the job. If 6 painters needed 77 hours to paint a 44,000 square foot office building, find how many hours would it take for 7 painters to paint a 34,000 square foot office building?
Round your answer to 2 decimal places, if necessary.

Question 79

The gravitational force between two heavenly bodies is inversely proportional to the square of the distance between the two bodies. If two planets that are 3,400 Gm apart share a gravitational attraction of 0.004343 N, what would the force between two planets of similar size that are 8400 Gm apart? Round the answer to 6 decimal places if necessary.

Question 80

What description best fits the power function, , pictured?

A) k is negative and p is negative and odd
B) k is positive and p is negative and odd
C) k is negative and p is negative and even
D) k is positive and p is negative and even