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

Question
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)
$$\displaystyle{y}{\left({x}\right)}=-\frac{{1}}{{16}}{\left({x}+{2}\right)}{\left({x}+{2}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}$$

### Relevant Questions

Write an equation for the polynomial graph:

y(x)=?
For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at $$\displaystyle{x}=−{3}$$ and triple zero $$\displaystyle{a}{t}{x}={0}$$. Passes through the point (1, 32).
Use your knowledge of the graphs of polynomial functions to make a rough sketch of the graph of $$\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}$$
Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. $$\displaystyle{y}={x}^{{{3}}}-{3}{x}^{{{2}}},{\left[-{2},{5}\right]}{b}{y}{\left[-{10},{10}\right]}$$
Graph the polynomial function. $$\displaystyle{f{{\left({x}\right)}}}=−{x}^{{{4}}}+{3}{x}^{{{3}}}−{x}+{1}$$
Describe the similarities between a) the lines $$\displaystyle{y}={x}{\quad\text{and}\quad}{y}=-{x}$$ and the graphs of other odd-degree polynomial functions b) the parabolas $$\displaystyle{y}={x}^{{{2}}}{\quad\text{and}\quad}{y}=-{x}^{{{2}}}$$ and the graphs of other even-degree polynomial functions
Make rough sketches of the graphs of each of the following polynomial functions. Be sure to label the x- and y- intercepts. $$\displaystyle{a}{)}{y}={x}{\left({2}{x}+{5}\right)}{\left({2}{x}-{7}\right)}$$ $$\displaystyle{b}{)}{y}={\left({15}-{2}{x}\right)}^{{{2}}}{\left({x}+{3}\right)}$$
a) Identify the parameters a, k, d, and c in the polynomial function $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{4}}}$$. b) State the domain and range, the vertex, and the equation of the axis of symmetry of the transformed function. c) Describe two possible orders in which the transformations can be applied to the graph of $$\displaystyle{y}={x}^{{{4}}}$$ to produce the graph of $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. d) Sketch graphs of the base function and the transformed function on the same set of axes.