91Ó°ÊÓ

The given function is $f\left(x\right)=(x-1{)}^2(x+3\left)\right(x+1)$.

For $x$-intercept, substitute $0$for $f\left(x\right)$and then solve for $x$.

$$\begin{gathered} 0=(x-1{)}^2(x+3\left)\right(x+1) \\ x=1,-1,-3 \end{gathered}$$

For $y$-intercept, substitute $0$for $y$and then solve for role="math" localid="1649266689167" $f\left(x\right)$.

$\begin{array}{rcl}f\left(x\right)& =& (0-1{)}^2(0+3\left)\right(0+1)\\ & =& 3\end{array}$

The zeros of the function are the x-intercepts.

Multiplicity is the number of times a certain factor appears in the function.

An odd multiplicity at a certain zero of the function means the graph crosses that point.

An even multiplicity at a certain zero of the function means the graph touches that point.

Find whether the graph crosses or touches the horizontal axis at the x-intercepts.

The zeros of $f\left(x\right)$are -3,-1 and 1 .

The zero -3 has a multiplicity of 1 , so the graph of $f\left(x\right)$crosses the horizontal axis at $x=-3$.

The zero -1 has a multiplicity of 1 , so the graph of $f\left(x\right)$crosses the horizontal axis at $x=-1$.

The zero 1 has a multiplicity of 2 , so the graph of $f\left(x\right)$touches the horizontal axis at $x=1$.

Identify the highest number of turning points on the graph of the function.

Since $f\left(x\right)$ has a degree of 4 , the graph has at most $4-1=3$ turning points.

By using the graphing utility, the graph of the given function is shown below:

There are 4 points where the graph changes from an increasing function to a decreasing one and vice versa.

Reading from the graph, the turning points are$\left(-2.28,-9.92\right)$,$\left(-0.22,3.23\right)$and$\left(1,0\right)$

The domain of the function is all real numbers, meaning$\left(-∞,+∞\right)$

The range of the function is all real numbers, meaningrole="math" localid="1649267499771" $[-10,+∞)$

The function $f$is increasing on $[-2.3,-0.2]\mathrm{and}[1,∞)$.

The function$f$is decreasing on $(-∞,-2.3]\text{and}[-0.2,1]$.

The y-intercept is 3 , the x-intercepts are 1,-3 and -1, while the end behavior model is $y=x^4$.

The turning points are (-2.3,-10),(-0.2,3.2) and (1,0), the function increasing on [-2.3,-0.2] and $[1,∞)$, decreasing on $(-∞,-2.3]$and $[-0.2,1]$while the domain is all real numbers which is $(-∞,+∞)$and range is set of real numbers starting from -10 which is $[-10,∞)$.