Question

Analyze each polynomial function f by following Steps 1 through 8 on page 192.

$f\left(x\right)=x^3-2.91x^2-7.668x+3.73248$

Short answer

x³

Step-by-step solution

Step 1. Given information.

The following polynomial function is given:

$f\left(x\right)=x^3-2.91x^2-7.668x+3.73248$

Step 2. Determine the end behaviour of the graph of the function.

Since the polynomial is of degree 3, the function behaves like$y=x^3$for large values of$\left|x\right|$.

Step 3. Graph the function using a graphing utility.

The graph of the function using a graphing utility is shown below:

Step 4. Use a graphing utility to approximate the x- and y-intercepts of the graph. 

The y-intercept is $f\left(0\right)=3.73248$and using a graphical calculator's ZERO function, the x-intercepts of the polynomial are at$-2.16,2.16,0.8$.

Step 5. Use a graphing utility to create a TABLE to find points on the graph around each x-intercept. 

Using a graphing utility, we have the following table:

Step 6. Approximate the turning points of the graph. 

Using the MAXIMUM and the MINIMUM functions of the graphing utility, we have the following turning points: $\left(-0.9,7,55\right),\left(2.84,-18.61\right)$after rounding the values to two decimal places.

Step 7. Use the information in Steps 2 through 6 to draw a complete graph of the function by hand. 

Using the information, the following graph can be obtained:

Step 8. Find the domain and the range of the function. 

The domain of the function is $\left(-\infty ,\infty \right)$and the range of the function is$\left(-\infty ,\infty \right)$.

Step 9. Use the graph to determine where the function is increasing and where it is decreasing. 

From the graph, we see that the function is increasing between $\left(-\infty ,-0.9\right)$and $\left(2.84,\infty \right)$. The function decreases between$\left(-0.9,2.84\right)$.