Question

Analyze each polynomial function.

$f\left(x\right)=-1.2x^4+0.5x^2-\sqrt{3}x+2$

Short answer

Consider the given function.

$f\left(x\right)=-1.2x^4+0.5x^2-\sqrt{3}x+2$

Step-by-step solution

Step 1. Given information.

Consider the given function.

$f\left(x\right)=-1.2x^4+0.5x^2-\sqrt{3}x+2$

Step 2. An Explanation for the polynomial.

The polynomial function fis of degree 4 and the graph behaves like, $y=-1.2x^4$.

The y-intercept is (0,2) and put 0 for f(x) to find x -intercept.

$\begin{array}{rcl}-1.2x^4+0.5x^2-\sqrt{3}x+2& =& 0\\ 1.2x^4-0.5x^2+\sqrt{3}x-2& =& 0\\ x& \approx & \hspace{0.17em}0.91336,\hspace{0.17em}-1.47191\end{array}$

The given function has two turning point

Values of x on each side of the y-intercepts: 

Make the table:

x	y
0	2
-1.472	0
0.913	0
-0.809	3.214

The given function has one turning point $\left(0.809,3.214\right)$.

The domain is the real numbers and the range is $(-\infty \hspace{0.17em},\hspace{0.17em}3.21445]$.

The increasing and decreasing interval is defined as,

$$\begin{gathered} \mathrm{Increasing}:\left(-\infty \hspace{0.17em}<x<-0.80894\right) \\ \mathrm{Decreasing}:\left(-0.80894<x<\infty \right)\hspace{0.17em} \end{gathered}$$

Step 4. The graph.

The graph of the given function is drawn as,