5.2 Power functions and polynomial functions  (Page 4/19)

For the function $f\left(x\right),$ the highest power of $x$ is 3, so the degree is 3. The leading term is the term containing that degree, $\mathrm{-4}{x}^3.$ The leading coefficient is the coefficient of that term, $\mathrm{-4.}$

For the function $g\left(t\right),$ the highest power of $t$ is $5,$ so the degree is $5.$ The leading term is the term containing that degree, $5t^5.$ The leading coefficient is the coefficient of that term, $5.$

For the function $h\left(p\right),$ the highest power of $p$ is $3,$ so the degree is $3.$ The leading term is the term containing that degree, $-p^3.$ The leading coefficient is the coefficient of that term, $\mathrm{-1.}$

As the input values $x$ get very large, the output values $f(x)$ increase without bound. As the input values $x$ get very small, the output values $f(x)$ decrease without bound. We can describe the end behavior symbolically by writing

In words, we could say that as $x$ values approach infinity, the function values approach infinity, and as $x$ values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Obtain the general form by expanding the given expression for $f\left(x\right).$

The general form is $f\left(x\right)=\mathrm{-3}{x}^4-9x^3+12x^2.$ The leading term is $\mathrm{-3}{x}^4;$ therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is