Chapter 4.1, Problem 70AYU

In Problems 77-80, write a polynomial function whose graph is shown (use the smallest degree possible).

To determine

To construct: A polynomial function of the following graph with minimum degree.

Answer to Problem 70AYU

$\text{The polynomial function }f\left(x\right)=\frac{-1}{4}\left(x+1\right)\left(x+1\right)\left(x-1\right)\left(x-4\right)$.

Explanation of Solution

Given:

The given zeros from the graph $=-1,1,4$.

There are 3 turning points. Therefore the least degree of the polynomial $=4$.

The graph crosses the $x\text{-axis}$ at $x=1,4$.

Therefore there will be odd multiplicity at $x=1,4$.

The graph touches the $x\text{-axis}$ at $x=-1$. Therefore there will be an even multiplicity at $x=-1$.

The polynomial function $f\left(x\right)=a\left(x+1\right)\left(x+1\right)\left(x-1\right)\left(x-4\right)=a(x^4-3x^3-5x^2+3x+4)$.

Where $a=$ the leading coefficient and is non-zero real number causing a stretch, compression or reflection and not affecting the $x\text{-intercept}$ of the graph.

The graph passes through the point $\left(3,8\right)$.

$8=a\left(3+1\right)\left(3+1\right)\left(3-1\right)\left(3-4\right)=8=a\left(-32\right)$

$a=\frac{8}{-32}=-\frac{1}{4}$

$f\left(x\right)=\frac{-1}{4}\left(x+1\right)\left(x+1\right)\left(x-1\right)\left(x-4\right)$