Chapter 5.8, Problem 37PPSE

Solution

To calculate : The equation of the quartic without using quartic regression calculator.

The required quartic function is $y=\frac{1}{8}{\left(x+2\right)}^2{\left(x-2\right)}^2$ .

Given information :

The graph given below is of a certain quartic polynomial in the ZDecimal window, which is $\left[-4.7,4.7\right]\text{ by }\left[-3.1,3.1\right]$ .

  

Calculation :

The graph represents the union of two parabolas with vertices $\left(-2,0\right)\text{ and }\left(2,0\right)$ .

The equation of parabola that opens up with vertex $\left(2,0\right)$ is $y={\left(x-2\right)}^2$ and the equation of parabola that opens up with vertex $\left(-2,0\right)$ is $y={\left(x+2\right)}^2$

To find the quartic function, multiply the two quadratic equations.

  $y={\left(x+2\right)}^2{\left(x-2\right)}^2$

From here, if the zeros of the quartic equation are $-2,2$ then $y=16$ .

But from the graph it can be seen that for the zeros $-2,2$ ; $y=2$ .

To make $y=2$ , divide the given model by $8$ . So,

  $y=\frac{{\left(x+2\right)}^2{\left(x-2\right)}^2}{8}$

Therefore, the required quartic function is $y=\frac{1}{8}{\left(x+2\right)}^2{\left(x-2\right)}^2$ .