5.2 Power Functions and Polynomial Functions

Use the property of exponents to simplify each of the following:

$\left(3 x^{2} y\right)^{4}$

If the expression is a polynomial, identify it as a monomial, binomial, trinomial, or other polynomial, and then find the degree of each polynomial. If it is not a polynomial, state this and give a reason why.

1. ⓐ $f \left(x\right) = x^{2}$
2. ⓑ $f \left(x\right) = 5 x^{4}$
3. ⓒ $f \left(x\right) = x^{2 / 3}$
4. ⓓ $f \left(x\right) = 3 x$
5. ⓔ $f \left(x\right) = - 5 x^{- 2}$

Which of the following functions are power functions?

Which functions are power functions?

$\begin{aligned}f \left(x\right) & = & 2 x \cdot 4 x^{3} \\ g \left(x\right) & = & - x^{5} + 5 x^{3} \\ h \left(x\right) & = & \frac{2 x^{5} - 1}{3 x^{2} + 4}\end{aligned}$

Describe in words and symbols the end behavior of $f \left(x\right) = - 5 x^{4} .$

Which of the following are polynomial functions?

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

Identify the degree, leading term, and leading coefficient of the polynomial $f \left(x\right) = 4 x^{2} - x^{6} + 2 x - 6.$

Given the function $f \left(x\right) = - 3 x^{2} \left(x - 1\right) \left(x + 4\right) ,$ express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Given the function $f \left(x\right) = 0.2 \left(x - 2\right) \left(x + 1\right) \left(x - 5\right) ,$ express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

Given the polynomial function $f \left(x\right) = 2 x^{3} - 6 x^{2} - 20 x ,$ determine the y- and x-intercepts.

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for $f \left(x\right) = - 3 x^{10} + 4 x^{7} - x^{4} + 2 x^{3} .$

Without graphing the function, determine the maximum number of x-intercepts and turning points for $f \left(x\right) = 108 - 13 x^{9} - 8 x^{4} + 14 x^{12} + 2 x^{3} .$

Given the function $f \left(x\right) = - 4 x \left(x + 3\right) \left(x - 4\right) ,$ determine the local behavior.

Given the function $f \left(x\right) = 0.2 \left(x - 2\right) \left(x + 1\right) \left(x - 5\right) ,$ determine the local behavior.

Explain the difference between the coefficient of a power function and its degree.

In general, explain the end behavior of a polynomial with odd degree if the leading coefficient is positive.

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As $x \rightarrow - \infty , f \left(x\right) \rightarrow - \infty$ and as $x \rightarrow \infty , f \left(x\right) \rightarrow - \infty .$

Identify the function as a power function, a polynomial function, or neither.

$f \left(x\right) = \left(x^{2}\right)^{3}$

Identify the function as a power function, a polynomial function, or neither.

$f \left(x\right) = \frac{x^{2}}{x^{2} - 1}$

Identify the function as a power function, a polynomial function, or neither.

$f \left(x\right) = 3^{x + 1}$

Find the degree and leading coefficient for the given polynomial.

$7 - 2 x^{2}$

Find the degree and leading coefficient for the given polynomial.

$x \left(4 - x^{2}\right) \left(2 x + 1\right)$

Determine the end behavior of the functions.

$f \left(x\right) = x^{4}$

Determine the end behavior of the functions.

$f \left(x\right) = - x^{4}$

Determine the end behavior of the functions.

$f \left(x\right) = - 2 x^{4} - 3 x^{2} + x - 1$

Determine the end behavior of the functions.

$f \left(x\right) = x^{2} \left(2 x^{3} - x + 1\right)$

Find the intercepts of the functions.

$f \left(t\right) = 2 \left(t - 1\right) \left(t + 2\right) \left(t - 3\right)$

Find the intercepts of the functions.

$f \left(x\right) = x^{4} - 16$

Find the intercepts of the functions.

$f \left(x\right) = x \left(x^{2} - 2 x - 8\right)$

Make a table to confirm the end behavior of the function.

$f \left(x\right) = x^{4} - 5 x^{2}$

Make a table to confirm the end behavior of the function.

$f \left(x\right) = \left(x - 1\right) \left(x - 2\right) \left(3 - x\right)$

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The $y -$ intercept is $\left(0 , - 4\right) .$ The $x -$ intercepts are $\left(- 2 , 0\right)$, $\left(2 , 0\right) .$ Degree is 2.

End behavior: as $x \rightarrow - \infty$, $f \left(x\right) \rightarrow \infty$; as $x \rightarrow \infty$, $f \left(x\right) \rightarrow \infty .$

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The $y -$ intercept is $\left(0 , 0\right) .$ The $x -$ intercepts are $\left(0 , 0\right)$, $\left(2 , 0\right) .$ Degree is 3.

End behavior: as $x \rightarrow - \infty$, $f \left(x\right) \rightarrow - \infty$, as $x \rightarrow \infty$, $f \left(x\right) \rightarrow \infty .$

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The $y -$ intercept is $\left(0 , 1\right) .$ There is no $x -$ intercept. Degree is 4.

End behavior: as $x \rightarrow - \infty$, $f \left(x\right) \rightarrow \infty$, as $x \rightarrow \infty$, $f \left(x\right) \rightarrow \infty .$

Use the written statements to construct a polynomial function that represents the required information.

A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of $m ,$ the number of minutes elapsed.

Use the written statements to construct a polynomial function that represents the required information.

An open box is to be constructed by cutting out square corners of $x -$ inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of $x .$