Question

Use the equation of the polynomial function f(x)=−2(x−6)x2−9 to complete the following.(a) State the degree and the leading coefficient.(b) Describe the end behavior of the graph of the function.(c) Support your answer by graphing the function.

Answer 1

we have the function

`f(x)=-2(x-6)(x^2-9)`

Part a

Apply the distributive property

`\begin{gathered} f(x)=-2(x^3-9x-6x^2+54) \\ f(x)=-2(x^3-6x^2-9x+54) \\ f(x)=-2x^3+12x^2+18x-108 \end{gathered}`

so

the degree of the polynomial function is 3 and the leading coefficient is -2

Part b

End behavior

we have that

Degree -----> 3 -----> is odd

leading coefficient ----> -2 -----> negative

therefore

f(x)→+∞, as x→−∞

f(x)→−∞, as x→+∞

Part c

graph

see the attached figure

Part b

the curve opens down to the right because the leading coefficient is negative. Because the polynomial is cubic the graph has end behavior in the opposite direction, so the other ends open up to the left