Question

Analyze each polynomial function using Steps 1 through 8.

$f\left(x\right)=x^3+8x^2+11x-20$

Short answer

The given function is analyzed and its graph is given as

Step-by-step solution

Step 1.  Determine the end behavior of the graph of the function.  

The given function$f\left(x\right)=x^3+8x^2+11x-20$ is of degree $3$.

The graph off behaves like $y=x^3$for large values of $\left|x\right|$.

Step 2. Find the x- and y-intercepts of the graph of the function.  

For $x=0$, the function value is given as

$$\begin{gathered} f\left(0\right)=0^3+8·0^2+11·0-20 \\ f\left(0\right)=-20 \end{gathered}$$

So they-intercept is $(0,-20)$.

Now set $f\left(x\right)=0$and solving for $x$we get

$\begin{array}{rcl}{x}^3+8x^2+11x-20& =& 0\\ x^3-x^2+9x^2-9x+20x-20& =& 0\\ x^2(x-1)+9x(x-1)+20(x-1)& =& 0\\ (x-1)(x^2+9x+20)& =& 0\\ (x-1)(x^2+5x+4x+20)& =& 0\\ (x-1)\left(x\right(x+5)+4(x+5\left)\right)& =& 0\\ (x-1)(x+5)(x+4)& =& 0\end{array}$

So the $x$intercepts are

$\begin{array}{rcl}x-1& =& 0\\ x& =& 1\end{array}$,

role="math" $\begin{array}{rcl}x+5& =& 0\\ x& =& -5\end{array}$

and

$\begin{array}{rcl}x+4& =& 0\\ x& =& -4\end{array}$

Step 3. Determine the zeros of the function and their multiplicity.  

The zeros of the function are $-5,-4,1$. Each zero is of multiplicity $1$, so the graph of the function crosses the $x$axis at each zero.

Step 4. Use a graphing utility to graph the function. 

The graph of the function can be given as

Step 5. Approximate the turning points of the graph.  

From the graph off , we see thatf has two turning points. Using MAXIMUM, one turning point is at $(-4.52,1.34)$, rounded to two decimal places. Using MINIMUM, the other turning point is at $(-0.81,-24.19)$, rounded to two decimal places.

Step 6. Use the information in Steps 1 to 5 to draw a complete graph of the function by hand.  

Plotting all the points and drawing a smooth curve we get

Step 7. Find the domain and the range of the function.  

The graph is defined for every value of x, so its domain is set of all real numbers.

The range of the function is also the set of all real numbers.

Step 8. Use the graph to determine where the function is increasing and where it is decreasing. 

Using the graph it can be seen that the graph is increasing in the interval $(-\infty ,-4.52)\cup (-0.81,\infty )$.

The graph is decreasing in the interval $(-4.52,-0.81)$.