5.2 Power functions and polynomial functions  (Page 7/19)

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Key equations

 general form of a polynomial function $f\left(x\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

Key concepts

• A power function is a variable base raised to a number power. See [link] .
• The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
• The end behavior depends on whether the power is even or odd. See [link] and [link] .
• A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See [link] .
• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See [link] .
• The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See [link] and [link] .
• A polynomial of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have at most $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ x- intercepts and at most $\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points. See [link] , [link] , [link] , [link] , and [link] .

Verbal

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

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

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

As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases without bound, so does $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without bound, so does $\text{\hspace{0.17em}}f\left(x\right).$

What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As $\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty \text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty .\text{\hspace{0.17em}}$

The polynomial function is of even degree and leading coefficient is negative.

Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

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

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

Power function

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

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

Neither

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

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

Neither

For the following exercises, find the degree and leading coefficient for the given polynomial.

$-3x{}^{4}$

$7-2{x}^{2}$

Degree = 2, Coefficient = –2

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

Degree =4, Coefficient = –2

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

For the following exercises, determine the end behavior of the functions.

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

$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

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

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

$\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty$

$f\left(x\right)=-{x}^{9}$

$\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty$

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

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

$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty$

$f\left(x\right)={\left(2-x\right)}^{7}$

For the following exercises, find the intercepts of the functions.

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

y -intercept is $\text{\hspace{0.17em}}\left(0,12\right),\text{\hspace{0.17em}}$ t -intercepts are

$g\left(n\right)=-2\left(3n-1\right)\left(2n+1\right)$

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

y -intercept is $\text{\hspace{0.17em}}\left(0,-16\right).\text{\hspace{0.17em}}$ x -intercepts are $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,0\right).$

$f\left(x\right)={x}^{3}+27$

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

y -intercept is $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ x -intercepts are $\text{\hspace{0.17em}}\left(0,0\right),\left(4,0\right),\text{\hspace{0.17em}}$ and

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

Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

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