Polynomial Functions follow the Standard Form:

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

The highest exponent present in a polynomial determines the **degree of the polynomial**.

$f\left(x\right)=3{x}^{2}+2x+5$ is a polynomial of degree 2

$f\left(x\right)=2{x}^{-1}+5$ is not a polynomial because it has a negative exponent

In the Graphs article, we looked at how to graph different types of polynomial functions (Line Graphs, quadratic, cubic, and quartic functions) but only based on the points where the curve crosses the x and y axes. However, as the behaviour of higher exponent functions is not as predictable as lines or parabolas, to get a more accurate representation of their curve, we need to use some key features.

## Key features of polynomial graphs

1. **Find the zeros:** The zeros of a function are the values of x that make the function equal to zero. They are also known as **x-intercepts**.

To find the zeros of a function, you need to set the function equal to zero and use whatever method required (factoring, division of Polynomials, Completing the Square or quadratic formula) to find the solutions for x. Please refer to the Polynomials article if you need a reminder of this.

After doing polynomial division and factoring of the polynomial function, we get the result $(x-1)(x+3)(x+4)=0$.

Based on this, the **zeros or x-intercepts** are:

$x=1$, $x=-3$ and $x=-4$

If a zero appears as part of the solution twice (it is repeated), then the curve of the function will touch the x-axis at that value of x, and then bounce off the x-axis changing its direction.

2. **Find the Turning Points (local maximum or minimum):** To find the highest point (local maximum) and the lowest point (local minimum) in a particular section of the curve where it changes direction, you should proceed as follows:

Find the derivative of the polynomial function using the power rule $f\text{'}\left(x\right)=n{x}^{n-1}$.

Make the function equal to zero to find the x-coordinates of the Turning Points. You can do this by factoring, Completing the Square or using the quadratic formula.

After this, you need to substitute the resulting values of x into the original function to find the y-coordinate of the turning points.

The derivative of $f\left(x\right)={x}^{3}+6{x}^{2}+5x-12$ is $f\text{'}\left(x\right)=3{x}^{2}+12x+5$

Now we need to find the x-coordinates of the turning points:

$3{x}^{2}+12x+5=0$

This polynomial cannot be factored, so let's use the quadratic formula

$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$

From the function, we can identify that $a=3$, $b=12$ and $c=5$

$x=\frac{-12\pm \sqrt{{12}^{2}-4\times 3\times 5}}{2\times 3}$

$x=\frac{-12\pm \sqrt{144-60}}{6}=\frac{-12\pm \sqrt{84}}{6}$ $\sqrt{84}=\sqrt{4x21}=\sqrt{4}\times \sqrt{21}=2\sqrt{21}$

$x=\frac{-12\pm 2\sqrt{21}}{6}$ Simplify by 2

$x=\frac{-6\pm \sqrt{21}}{3}$

We have two solutions, which are the x-coordinates of the turning points:

$x=\frac{-6+\sqrt{21}}{3}=-0.472$

$x=\frac{-6-\sqrt{21}}{3}=-3.528$

- Now we substitute the resulting values of x of into the original function to find the y-coordinate of the turning points:

$f(-0.472)={(-0.472)}^{3}+6{(-0.472)}^{2}+5(-0.472)-12$

$f(-0.472)=-13.128$

$f(-3.528)={(-3.528)}^{3}+6{(-3.528)}^{2}+5(-3.528)-12$

$f(-3.528)=1.128$

The **turning points** are:

Local maximum = $(-3.528,1.128)$

Local minimum = $(-0.472,-13.128)$

3. **Find the y-intercept:** Substitute x = 0 in the original polynomial function. The result will be the y-coordinate where the curve crosses the y-axis.

$f\left(x\right)={x}^{3}+6{x}^{2}+5x-12$

$f\left(0\right)={0}^{3}+6{\left(0\right)}^{2}+5\left(0\right)-12$

$f\left(0\right)=-12$

The point where the curve of the function crosses the y-axis is (0, -12)

4. **End Behaviour:** The curves of polynomials that have a degree of 2 or more are continuous and smooth lines that can have maximum or minimum points where they change direction in the middle section of the curve, and on either end of the curve they tend to go towards positive or negative infinity.

**How do you determine the end behaviour of a function?**

**Leading Coefficient Test:** The **leading term of a polynomial **is the term with the highest exponent. You will need to look at whether its exponent is even or odd and the sign of its coefficient to help you determine the end behaviour of the curve.

**Odd function**(i.e. ${x}^{3},{x}^{5},{x}^{7}$)

a) **Positive leading coefficient:** In this case, the function will point downwards on the left and point up on the right end of the curve.

b) **Negative leading coefficient:** In this case, the function will point up on the left and point down on the right end of the curve.

**Even function**(i.e. ${x}^{2},{x}^{4},{x}^{6}$)

a) **Positive leading coefficient:** In this case, the function will point upwards on both ends of the curve.

b) **Negative leading coefficient:** In this case, the function will point downwards on both ends of the curve.

Odd function | Even function | |||||||

Sign of the Leading Coefficient | Positive | Negatives | Positive | Negatives | ||||

End Behaviour | Left | Right | Left | Right | Left | Right | Left | Right |

↓ | ↑ | ↑ | ↓ | ↑ | ↑ | ↓ | ↓ |

$f\left(x\right)={x}^{3}+6{x}^{2}+5x-12$

The leading term of the polynomial function is ${x}^{3}$, which means that it is an **odd function **with a **positive leading coefficient. **Therefore, the end behaviour of the curve will be like this:

Left | Right |

↓ | ↑ |

5. **Sketch the curve of the function**.

## What are the different types of polynomial graphs?

There are different types of polynomial graphs according to their degree.

Notice that the degree of a polynomial matches the Number of direction changes in their graph and the Number of zeros or x-intercepts.

Degree 1 - Linear | Degree 2 - Quadratic |

Degree 3 - Cubic | Degree 4 - Quartic |

Degree 5 - Quintic | Degree 6 |

## How do you find the equation of a polynomial function from its graph?

If you are given the graph of a polynomial function, you can find the equation of the polynomial function by following these steps:

Identify the zeros or x-intercepts (values of x where the curve crosses or touches the x-axis).

Write the Factors of the function using the zeros identified (make sure that you change the sign of the zeros when you write them as factors). For example, if b is a root, then$(x-b)$ is a factor of the function.

Any repeated Factors can be written as ${(x\pm b)}^{2}$.

Find the value of the stretch factor $\left(a\right)$ using the y-intercept.

Find the equation of the polynomial function represented by the following graph:

1. The zeros or x-intercepts are:

$x=-4$, $x=-3$ and $x=1$

2. The factors are: $(x+4)(x+3)(x-1)$

3. $f\left(x\right)=a(x+4)(x+3)(x-1)$ $a=?$

The y-intercept is (0, -12), so you need to substitute those values in $f\left(x\right)$ to find the value of the stretch factor $\left(a\right)$.

$-12=a(0+4)(0+3)(0-1)$

$-12=a\left(\right(4\left)\right(3\left)\right(-1\left)\right)$

$-12=-12a$

$a=\frac{-12}{-12}=1$

$a=1$, therefore the equation of the polynomial function is:

$f\left(x\right)=(x+4)(x+3)(x-1)$

You can leave it like this, or expand brackets and combine like terms to get the Standard Form of the polynomial function, like this:

$f\left(x\right)=(x+4)(x+3)(x-1)$ expand the first two brackets first

$=({x}^{2}+3x+4x+12)(x-1)$

$=({x}^{2}+7x+12)(x-1)$

$={x}^{3}+7{x}^{2}+12x-{x}^{2}-7x-12$

$f\left(x\right)={x}^{3}+6{x}^{2}+5x-12$

## Polynomial Graphs - Key takeaways

Polynomial graphs are graphical representations of polynomial functions.

Some key features of polynomial graphs are the number of zeros or x-intercepts, the repeated zeros, the turning points, the y-intercept, the type of function (odd or even), the sign of the leading coefficient, and the end behaviour of the curve.

There are different types of polynomial graphs according to their degree.

The degree of a polynomial matches the number of direction changes in their graph and the number of zeros or x-intercepts.

It is possible to find the equation of a polynomial function from its graph.

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##### Frequently Asked Questions about Polynomial Graphs

What is a polynomial graph?

Polynomial graphs are graphical representations of polynomial functions.

How do you graph polynomial functions?

To graph polynomial functions follow these steps:

- Find the zeros using whatever method required (factoring, division of polynomials, completing the square or quadratic formula).
- Find the turning points (local maximum or minimum).
- Find the y-intercept.
- Carry out the leading coefficient test to find the end behaviour of the polynomial function.
- Sketch the function.

How do you find the equation of a polynomial graph?

To find the equation of the polynomial function from its graph, follow these steps:

- Identify the zeros or x-intercepts (values of x where the curve crosses or touches the x-axis.
- Write the factors of the function using the zeros identified (make sure that you change the sign of the zeros when you write them as factors).
- Any repeated factors can be written as (x± b)².
- Find the value of the stretch factor (a) using the y-intercept.

What are examples of polynomial graphs?

Examples of polynomial graphs include: degree 1 - linear, degree 2 - quadratic, degree 3 - cubic, and degree 4 - quartic.

What are some key features of polynomial graphs?

Some key features of polynomial graphs are: the number of zeros or x-intercepts, the repeated zeros, the turning points, the y-intercept, the type of function (odd or even), the sign of the leading coefficient, and the end behaviour of the curve.

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