# Write an equation for the polynomial graph:01510102861.jpgy(x)=?

Polynomial graphs

Write an equation for the polynomial graph:

$$y(x)=?$$

2021-03-08
$$\displaystyle{x}_{{1}}=-{2},{x}_{{2}}=-{2},{x}_{{3}}={2},{x}_{{4}}={4}$$
then the factors of y(x)
$$\displaystyle{\left({x}-{x}_{{1}}\right)}{\left({x}-{x}_{{2}}\right)}{\left({x}-{x}_{{3}}\right)}{\left({x}-{x}_{{4}}\right)}$$
$$\displaystyle{\left({x}-{\left(-{2}\right)}\right)}{\left({x}-{\left(-{2}\right)}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}$$
$$\displaystyle{\left({x}+{2}\right)}{\left({x}+{2}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}$$
y intercept is (0,-2)
put y and x values for k calculation
$$\displaystyle-{2}={k}{\left({0}+{2}\right)}{\left({0}+{2}\right)}{\left({0}-{2}\right)}{\left({0}-{4}\right)}$$
$$\displaystyle-{2}={k}{32}$$
$$\displaystyle{k}=-\frac{{2}}{{32}}=-\frac{{1}}{{16}}$$
$$\displaystyle{k}=-\frac{{1}}{{6}}$$
Now put the value of k in y(x)
$$\displaystyle{y}{\left({x}\right)}=-\frac{{1}}{{16}}{\left({x}+{2}\right)}{\left({x}+{2}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}$$