Chapter 4.1, Problem 104AYU

To determine

To analyze: The polynomial function.

Answer to Problem 104AYU

$\text{The polynomial function }f\left(x\right)=4x^3+10x^2-4x-10\text{ is analyzed}$.

Explanation of Solution

Given:

$f\left(x\right)=4x^3+10x^2-4x-10$

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

$f\left(x\right)=4x^3+10x^2-4x-10$

Factor

$f\left(x\right)=4x^3+10x^2-4x-10=\left(x+1\right)\left(x-1\right)\left(2x+5\right)$

The polynomial function $f$ is of degree 3. The graph of $f$ behaves like $y=x^3$ for large values of $\left|x\right|$.

Step 2: Find $x$ and $y\text{-intercepts}$ of the graph of the function.

The $y\text{-intercept }=f\left(0\right)=-10$.

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

$f\left(x\right)=\left(x+1\right)\left(x-1\right)\left(2x+5\right)$

$x=-1,1,-\frac{5}{2}$

The $x\text{-intercept }=x=-1,1,-\frac{5}{2}$.

Step 3: Determine the zeros of the function and their multiplicity. Using this information determine whether the graph crosses or touches the $X\text{-axis}$ at each $x\text{-intercept}$.

The zeros of $f=-1,1,-\frac{5}{2}$.

The zero $-1$ has odd multiplicity therefore the graph of $f$ crosses the $X\text{-axis}$.

The zero $1$ has odd multiplicity therefore the graph of $f$ crosses the $X\text{-axis}$.

The zero $-\frac{5}{2}$ has odd multiplicity therefore the graph of $f$ crosses the $X\text{-axis}$.

Step 4: Using the graphing utility to graph the function.

$y_1=4x^3+10x^2-4x-10$

Step 5: Approximate the turning point of the graph.

From the graph of $f$, we see $f$ has 2 turning points.

Using the graph maximum turning point $=\left(-2.25,6\right)$.

Using the graph minimum turning point $=\left(0.1,-10.5\right)$.

Step 6: Redraw the graph.

The graph passes through $\left(1,0\right)\left(-2.5,0\right)\left(-1,0\right)$.

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 $\left(-\infty ,-2.25\right]$ and $\left[0,\infty \right)$.

Based on the graph, $f$ is decreasing $\left[-2.25,0\right]$.