# Function Graphs

## Function Graphs Revision

**Function Graphs**

You have so far seen graphs of straight lines, quadratics and cubics. Now it is time to see some graphs of more complicated functions, such as graphs of **y = kx^n** and **quartics**. As before, you will only need to draw a rough sketch of these graphs.

**Graphs of y = kx^n**

**Graphs of y = kx^n** have the same basic shape, however the values of k and n affect how they look.

Here are some examples:

**n positive and even**

The graphs are either:

**u-shaped**if k is**positive****n-shaped**if k is**negative**

**n positive and odd**

The graphs either go:

**bottom-left to top-right**if k is**positive****top-left to bottom-right**if k is**negative**

(the graphs go ‘corner-to-corner’)

**n negative and even**

The graphs are mirror images in the y-axis, and are:

**above the x-axis**if k is**positive****below the x-axis**if k is**negative**

**n negative and odd**

The parts of the graph are in diagonally opposite quadrants, and are:

**in the bottom-left and top-right quadrants**if k is**positive****in the top-left and bottom-right quadrants**if k is**negative**

**Note:** an asymptote is a line that the curve gets infinitely close to, but **never touches**. The third and fourth graphs both have asymptotes at x=0 and y=0.

**Quartics**

A **quartic** is a polynomial with an x^4 term as the highest power. To sketch a quartic, you will need to find where the curve crosses or touches the x-axis – the expression will be usually factorised which will make it easier to find these values.

**Quartics** with positive x^4 coefficients are **positive** for very positive and very negative x-values. Quartics with negative x^4 coefficients are **negative** for very positive and very negative x-values.

**Example:** Sketch the graph of f(x) = x^2(x-1)(x+2)

Let f(x)=0 to find the points where the curve crosses the x-axis:

x^2(x-1)(x+2) = 0

x^2 is a double root, so the graph touches the x-axis at \textcolor{red}{0}

The curve crosses the x-axis at \textcolor{red}{1} and \textcolor{red}{-2}

Substitute in x=0 to find where the curve crosses the y-axis:

y = 0^2(0-1)(0-2) = \textcolor{red}{0}

The coefficient of x^4 is **positive**, so the curve will be positive for very positive and very negative x-values.

Hence, we have enough information to sketch the graph.

## Function Graphs Example Questions

**Question 1:** Sketch the graph of y=-2x^5, labelling any points of intersection with the axes.

**[2 marks]**

The value of n is positive and odd (5), and the value of k is negative (-2).

Hence, the graph will have a corner-to-corner shape from top-left to bottom-right.

The graph will pass through the origin.

Therefore, the graph will look like this:

**Question 2:** Sketch the graph of y=\dfrac{4}{x^{3}}, labelling any points of intersection with the axes.

**[2 marks]**

The value of n is negative and odd (-3), and the value of k is positive (4).

Hence, the graph will be in the bottom-left and top-right corner.

Therefore, the graph will look like this:

**Question 3:** Sketch the graph of y=-(x-1)^2 (x+1)^2, labelling any points of intersection with the axes.

**[3 marks]**

The coefficient of x^4 is negative, therefore the curve will be negative for very positive and very negative x-values.

Find the points where the curve crosses the x-axis:

-(x-1)^2 (x+1)^2 = 0

(x-1)^2 and (x+1)^2 are both double roots, therefore the curve will touch the x-axis at x=1 and x=-1

Find the point where the curve intersects the y-axis:

y = -(0-1)^2 (0+1)^2 = -1

Therefore, we have enough information to sketch the graph: