# How do you graph a polynomial function?

Aug 12, 2018

This is quite a broad question.
Tips below.

#### Explanation:

Let $f \left(x\right)$ be a polynomial of n^(th degree with real coefficients.

To plot the graph of $f \left(x\right)$ the following points are useful.

(i) Find the real zeros of $f \left(x\right)$, if any.

Set $f \left(x\right) = 0$ and solve for $x$.
The real zeros are points on the $x -$axis.

(ii) Find the $y -$intercept.
Find the point $f \left(0\right)$. This is the intercept on the $y -$axis.

(iii) Find the turning points of $f \left(x\right)$, if any.

Set $f ' \left(x\right) = 0$ and solve for $x$. (Say, $\overline{x}$)

Then,
where $f ' ' \left({x}_{i}\right) < 0 \to f \left({x}_{i}\right)$ is a local maximum value.
where $f ' ' \left({x}_{i}\right) > 0 \to f \left({x}_{i}\right)$ is a local minimum value.
where $f ' ' \left({x}_{i}\right) = 0 \to f \left({x}_{i}\right)$ is an inflection point.

(iv) Plot points.

Outside of the above simply compute $f \left({x}_{j}\right)$ and plot points $\left({x}_{j} , f \left({x}_{j}\right)\right)$ as necessary to complete the graph.

I hope this helps.