# Graphing Polynomial Functions

## Key Questions

• A polynomial function in standard form must look like:

$f \left(x\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + {a}_{n - 2} {x}^{n - 2} + \ldots + {a}_{2} {x}^{2} + {a}_{1} x + {a}_{0}$

where $n \in \mathbb{N}$ and ${a}_{i} \in \mathbb{Z} , i \in \left\{0 , 1 , 2 , \ldots , n\right\}$

If you are not familiar with the notation, $\mathbb{N}$ is the set of natural number, and $\mathbb{Z}$ is the set of integers.

Examples of polynomial functions in standard form:

$f \left(x\right) = - 4 {x}^{5} + 7 {x}^{3} - 6 {x}^{2} + 4$
$g \left(x\right) = 3 {x}^{4} - 4 {x}^{3} + 6 x - 9$

Examples of non polynomial functions:

$h \left(x\right) = 4 {x}^{7} - 3 {x}^{5} + \sqrt{2 x} - 5$
$j \left(x\right) = - 7 {x}^{3} - 2 {x}^{2} + \frac{5}{{x}^{3}}$
$k \left(x\right) = 4.5 {x}^{5} - 3 {x}^{2}$

I'll leave it to you to figure out why they are non polynomial functions.

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.