**On this Page**:Turning Points

Leading Terms, Coefficients

Leading Coefficient Test

Multiplicity of a Zero

Graphing Steps & Example

Here we look at how to approach graphing polynomial functions examples, where we want to try to make as accurate a sketch as we can of a polynomial graph on an appropriate axis.

Some examples of graphing quadratic graphs which are polynomials of degree 2 can be seen *here*.

When a polynomial is of larger degree though, sometimes we need to do a bit more work for some more information on what the shape of the graph should be.

Here are two examples of what a polynomial graph can look like.

Polynomial graphs are smooth graphs and have no holes or breaks in them.

### Turning Points:

The curves on polynomial graphs are called the ‘turning points’.

A polynomial of degree

*n*,

has at most

*n*− 1 turning points on its graph.

So the graph of a polynomial such as f(x) = x^3 + 5x^2 \space {\text{–}} \space 2x + 1,

will have at most 2 turning points.

### Leading Terms and Leading Coefficients:

With learning how to deal with graphing polynomial functions examples it’s important to be clear about the leading term and leading coefficient is in a polynomial.

For a polynomial, P(x) = a_{n}x^n + a_{n \space {\text{–}} \space 1}x^{n \space {\text{–}} \space 1} + … + a_{1}x + a_0.

a_n is the leading coefficient,

a_{n}x^n is the leading term.

So with g(x) = 2x^3 + x^2 \space {\text{–}} \space 2x + 5,

the leading coefficient is 2, the leading term is 2x^3.

### Leading Coefficient Test:

Something called the ‘leading coefficient test’ can help us establish the behaviour of the polynomial graph at each end.

As stated, for a polynomial P(x) = a_{n}x^n + a_{n \space {\text{–}} \space 1}x^{n \space {\text{–}} \space 1} + … + a_{1}x + a_0,

the leading term is a_{n}.

There are 4 cases we can have with the leading term, and what it can tell us.

**1)**a_{n} > 0 \space\space , \space\space n even. The graph will increase with no bound to the left and the right.

**2)**a_{n} > 0 \space\space , \space\space n odd. The graph will decrease to the left, and increase on the right with no bound.

**3)**a_{n} < 0 \space\space , \space\space n even. The graph will decrease with no bound to the left and the right.

**4)**a_{n} < 0 \space\space , \space\space n odd. The graph will increase on the left, and decrease to the right with no bound.

### Multiplicity of a Zero:

The multiplicity of a zero is the number of times it appears in the factored polynomial form.

For (x \space {\text{–}} \space 5)^3(x + 1)^2 = 0.

The zeros are 5 and {\text{-}}1.

Of which, 5 has multiplicity 3, and {\text{-}}1 has multiplicity 2.

So:

When (x \space {\text{–}} \space t)^m is a factor,

*t*is a zero.

If

*m*is even, the x-intercept at x = t will only touch the x-axis, but not cross it.

If

*m*is odd, the x-intercept at x = t will cross the x-axis.

Also if

*m*is greater than 1, the curve of the polynomial graph will flatten out at the zero.

## Graphing Polynomial Functions Examples

Steps

So in light of what we’ve seen so far on this page, to make a sketch of the graph of a polynomial function there are some steps to take.

**A)**Find the zeros of the polynomial and their multiplicity.

**B)**Use the leading coefficient test to establish the graph end behaviour.

**C)**Find the y-intercept, (0,P(0)).

**D)**Establish the maximum number of turning points there can be, n \space {\text{–}} \space 1.

**E)**Plot some extra points for some more information of graph shape between the zeros.

Part E) will become more clear in an example.

__Example__

*(1.1)*Sketch the graph of the polynomial f(x) = x^4 \space {\text{–}} \space 3x^3 \space {\text{–}} \space 9x^2 + 23x \space {\text{–}} \space 12.

*Solution*Often we are required to find the factors and zeros, but as to concentrate on sketching a graph the polynomial here, we will state them.

f(x) = x^4 \space {\text{–}} \space 3x^3 \space {\text{–}} \space 9x^2 + 23x \space {\text{–}} \space 12 \space = \space (x \space {\text{–}} \space 1)^{\tt{2}}(x \space {\text{–}} \space 4)(x + 3)

(x \space {\text{–}} \space 1)^{\tt{2}}(x \space {\text{–}} \space 4)(x + 3) \space = \space 0 => x = 1 \space , \space x = 4 \space , \space x = {\text{-}}3

Of the zeros, 1 has multiplicity 2, while 4 and {\text{-}}3 have multiplicity 1.

So the graph will cross the x-axis at –

**3**and

**4**, but only touch at

**1**.

The leading coefficient is

**1**, which is

**> 0**, and n =

**4**.

So the graph will increase with no bound past both the left and right ends.

For the y-intercept. f(0) = {\text{-}}12 =>

**(0,-12)**

As n = 4, there can be at most 3 turning points, if our graph has any more we have done something wrong somewhere.

With what we know thus far, we can make a start at a sketch of the polynomial graph, on a suitable axis.

Now we look to establish some extra information about graph behaviour between the zeros or other point son the x-axis.

Between 1 and 4. f(3) = {\text{-}}24 \space , \space f(2) = {\text{-}}10

So there is a turning point between 1 and 4.

With n \space {\text{–}} \space 1 = 3, there can only be one more turning point.

Between -3 and 0. f({\text{-}}2) = {\text{-}}54 \space , \space f({\text{-}}\frac{3}{2}) = {\text{-}}51.56

There is a turning point between -3 and 0.

In calculus we can work out more exact turning points and make more exact drawings of polynomial graphs when dealing with graphing polynomial functions examples.

But for a rough sketch of the graph, what we have above gives us a good idea and is fairly accurate.

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