Graphs of Functions

A function is a rule that links each input value to exactly one output value Functions can be represented visually using graphs.

Graphs are drawn on a set of axes, where the axis represents the input and the axis represents the output.

Each point on a graph is written as a coordinate showing a pair of values that satisfy the rule in the function.

Different types of functions produce different graph shapes. At GCSE, we need to be able to recognise, interpret, and sketch these graphs, and use them to solve problems.

Linear Graphs

Linear equations are in the form

Linear equations are represented by straight lines, where is the gradient of the line and is the intercept, the point where the line crosses the axis.

Here is the graph of when (positive gradient) and (positive intercept).

Similarly, here is the graph of when (negative gradient) and

Quadratic Graphs

Quadratic equations are in the form

where

The graph of a quadratic is called a parabola. The value of is the intercept. The graph has a turning point, which is a minimum if or a maximum if All quadratics are symmetrical at the turning points.

Here is the graph of when and (intersects the negative axis).

Similarly, here is the graph of when and

Cubic Graphs

Cubic equations are in the form

where

The graph of a cubic is a smooth S-shaped curve, and the value of is the intercept. Below is the graph of when In this case, the value of is so the intercept is at the origin.

Similarly, here is the graph of when

A typical cubic graph, with equation has two turning points. Here is a cubic graph where

Similarly, here is a typical cubic graph where

Reciprocal Graphs

Reciprocal graphs show an inverse relationship between and where as increases, decreases, and vice versa. The equations of these graphs are of the form

where is a constant and The and axes are asymptotes, meaning the graph gets very close to them but never meets or crosses them. Here is the graph of when

Similarly, here is the graph of when

Exponential Graphs

An exponential graph shows a relationship where a quantity increases or decreases by a constant factor. Exponential functions can be used to model real-life situations such as population growth or radioactive decay. The equations of these graphs are in the form

where and

Notice how is in the power here (also known as the exponent, hence why we call them exponential graphs).

To find the intercept, we must substitute which, in this case, gives us:

Recall, from the index rules, that if we raise anything to the power of we get So,

Therefore, the intercept is

The axis is an asymptote, meaning that the graph gets very close to it but never meets or crosses it.

Here is the graph of when and

This graph is showing exponential growth. As increases, also increases.

Similarly, here is the graph of when and

The graph is showing exponential decay. As increases, decreases. The important condition is When a number between and is raised to higher powers, the result of becomes smaller and smaller.

Take the graph of as an example. Let’s substitute some values of to see what happens.

We can see that as the power (the value of increases, the value of decreases, so the graph trends downwards towards (the axis) as increases but it never reaches it.

This decay graph can also be linked to exponential growth. For example, using our index rules, we can rewrite the graph of as

We can get the graph of by reflecting the graph of in the axis. Therefore, we can think of exponential decay graph as the reflection in the axis of the exponential growth graph.

If the graph is reflected in the axis, so all values become negative while the values stay the same.

Please read our study note on Transformations of Functions for more on these.