Chapter 4.1, Problem 96AYU

To determine

To analyze: The polynomial function.

Answer to Problem 96AYU

$\text{The polynomial function }f\left(x\right)=x^4-18.5x^2+50.2619\text{ is analyzed}$.

Explanation of Solution

Given:

$f\left(x\right)=x^4-18.5x^2+50.2619$

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

Analyze the graph of the polynomial function.

$f\left(x\right)=x^4-18.5x^2+50.2619$

The polynomial function $f$ is of degree 4. The graph of $f$ behaves like $y=x^4$ for large values of $\left|x\right|$.

Step 2: Graph the function using a graphing utility.

$Y_1=x^4-18.5x^2+50.2619$

Step 3: Use a graphic utility to approximate the $x$ and $y\text{-intercepts}$ of the graph.

The $y\text{-intercept }=f\left(0\right)=50.2619$.

Using graphic utility find the lone $x\text{-intercept }=-2,-1,1,2$.

Step 4: Using the graphic utility to find the points on the graph around each $x\text{-intercept}$.

The points $\left(-2,0\right)\left(-1,0\right)\left(1,0\right)\left(2,0\right)\left(-2,-35\right)\left(2,35\right)$ are on the graph.

Step 5: Approximate the turning point of the graph.

From the graph of $f$, we see $f$ has 3 turning points.

Using the graph maximum turning point $=\left(0,48.5\right)$ rounded to two decimal places.

Using the graph minimum turning point $=\left(-2,-35\right)$ and $\left(2,35\right)$ rounded to two decimal places.

Step 6: Redraw the graph.

The graph passes through $\left(-2,0\right)\left(-1,0\right)\left(1,0\right)\left(2,0\right)\left(-2,-35\right)\left(2,35\right)$.

The end behavior of the graph $=y=x^4$.

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

The domain and range is the set of all real numbers.

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

Based on the graph, $f$ is increasing on the intervals $\left[-2,0\right]$ and $\left[2,\infty \right)$.

Based on the graph, $f$ is decreasing $\left(\infty ,-2\right]$ and $\left[0,2\right]$.