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general form of a polynomial function | $$f(x)={a}_{n}{x}^{n}+\mathrm{...}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$$ |
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 $\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f(x)\to -\infty \text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to -\infty .\text{\hspace{0.17em}}$
The polynomial function is of even degree and leading coefficient is negative.
For the following exercises, identify the function as a power function, a polynomial function, or neither.
$f(x)={x}^{5}$
$f(x)=x-{x}^{4}$
$f(x)=2x\left(x+2\right){\left(x-1\right)}^{2}$
For the following exercises, find the degree and leading coefficient for the given polynomial.
$-3x{}^{4}$
$-2{x}^{2}-3{x}^{5}+x-6$
${x}^{2}{\left(2x-3\right)}^{2}$
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(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f(x)\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(x)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to -\infty $
$f\left(x\right)=-{x}^{9}$
$f(x)=-2{x}^{4}-3{x}^{2}+x-1$
$\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to -\infty $
$f(x)=3{x}^{2}+x-2$
$f(x)={x}^{2}\left(2{x}^{3}-x+1\right)$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f(x)\to -\infty $
$f(x)={(2-x)}^{7}$
For the following exercises, find the intercepts of the functions.
$f\left(t\right)=2\left(t-1\right)\left(t+2\right)(t-3)$
y -intercept is $\text{\hspace{0.17em}}(0,12),\text{\hspace{0.17em}}$ t -intercepts are $\text{\hspace{0.17em}}(1,0);(\u20132,0);\text{and}(3,0).$
$g\left(n\right)=\mathrm{-2}\left(3n-1\right)(2n+1)$
$f(x)={x}^{4}-16$
y -intercept is $\text{\hspace{0.17em}}(0,-16).\text{\hspace{0.17em}}$ x -intercepts are $\text{\hspace{0.17em}}(2,0)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(-2,0).$
$f(x)={x}^{3}+27$
$f(x)=x\left({x}^{2}-2x-8\right)$
y -intercept is $\text{\hspace{0.17em}}(0,0).\text{\hspace{0.17em}}$ x -intercepts are $\text{\hspace{0.17em}}(0,0),(4,0),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,0\right).$
$f(x)=(x+3)\left(4{x}^{2}-1\right)$
For the following exercises, determine the least possible degree of the polynomial function shown.
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