Question

How do you graph a polynomial function?

Answer 1

This is quite a broad question.
Tips below.

Explanation:

Let `f(x)` be a polynomial of `n^(th` degree with real coefficients.

To plot the graph of `f(x)` the following points are useful.

(i) Find the real zeros of `f(x)`, if any.

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

(ii) Find the `y-`intercept.
Find the point `f(0)`. This is the intercept on the `y-`axis.

(iii) Find the turning points of `f(x)`, if any.

Set `f'(x) = 0` and solve for `x`. (Say, `barx`)

Then,
where `f''(x_i)<0 -> f(x_i)` is a local maximum value.
where `f''(x_i)>0 -> f(x_i)` is a local minimum value.
where `f''(x_i)=0 -> f(x_i)` is an inflection point.

(iv) Plot points.

Outside of the above simply compute `f(x_j)` and plot points `(x_j, f(x_j))` as necessary to complete the graph.

I hope this helps.