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In this section, you will:
  • Identify power functions.
  • Identify end behavior of power functions.
  • Identify polynomial functions.
  • Identify the degree and leading coefficient of polynomial functions.
Three birds on a cliff with the sun rising in the background.
(credit: Jason Bay, Flickr)

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in [link] .

Year 2009 2010 2011 2012 2013
Bird Population 800 897 992 1 , 083 1 , 169

The population can be estimated using the function P ( t ) = 0.3 t 3 + 97 t + 800 , where P ( t ) represents the bird population on the island t years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.

Identifying power functions

In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)

As an example, consider functions for area or volume. The function for the area of a circle with radius r is

A ( r ) = π r 2

and the function for the volume of a sphere with radius r is

V ( r ) = 4 3 π r 3

Both of these are examples of power functions because they consist of a coefficient, π or 4 3 π , multiplied by a variable r raised to a power.

Power function

A power function    is a function that can be represented in the form

f ( x ) = k x p

where k and p are real numbers, and k is known as the coefficient    .

Is f ( x ) = 2 x a power function?

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.

Identifying power functions

Which of the following functions are power functions?

f ( x ) = 1 Constant function f ( x ) = x Identify function f ( x ) = x 2 Quadratic  function f ( x ) = x 3 Cubic function f ( x ) = 1 x   Reciprocal function f ( x ) = 1 x 2 Reciprocal squared function f ( x ) = x Square root function f ( x ) = x 3 Cube root function

All of the listed functions are power functions.

The constant and identity functions are power functions because they can be written as f ( x ) = x 0 and f ( x ) = x 1 respectively.

The quadratic and cubic functions are power functions with whole number powers f ( x ) = x 2 and f ( x ) = x 3 .

The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as f ( x ) = x 1 and f ( x ) = x 2 .

The square and cube root functions are power functions with fractional powers because they can be written as f ( x ) = x 1/2 or f ( x ) = x 1/3 .

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Which functions are power functions?

f ( x ) = 2 x 2 4 x 3 g ( x ) = x 5 + 5 x 3 4 x h ( x ) = 2 x 5 1 3 x 2 + 4

f ( x ) is a power function because it can be written as f ( x ) = 8 x 5 . The other functions are not power functions.

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Identifying end behavior of power functions

[link] shows the graphs of f ( x ) = x 2 , g ( x ) = x 4 and and h ( x ) = x 6 , which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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