Graphs of Exponential Functions

What makes a function exponential?

At Higher tier, you need to recognise, sketch and interpret functions of the form

$y=k^x,$

where the base $k$ is a positive constant. The variable $x$ is in the exponent: this distinguishes an exponential function from a power such as $y=x^2$.

The output changes by the same multiplicative factor whenever $x$ increases by 1. For example, in $y=2^x$, moving one unit to the right doubles the value of $y$:

$\ldots,\frac14,\frac12,1,2,4,8,\ldots$

This repeated multiplication, rather than repeated addition, creates the characteristic curved graph.

Features shared by all the graphs

Every graph $y=k^x$ with $k>0$ passes through $(0,1)$ because

$k^0=1.$

Its $y$-values are always positive: a positive base raised to any real power gives a positive result. The graph therefore lies entirely above the $x$-axis and has no $x$-intercept.

Negative powers produce reciprocals. For instance,

$2^{-1}=\frac12,\qquad 2^{-2}=\frac14.$

This explains why a growth graph has small positive values to the left of the $y$-axis. Its curve approaches $y=0$ but never reaches it, so the $x$-axis is a horizontal asymptote.

How the value of $k$ changes the graph

There are three cases:

• If $k>1$, the graph shows exponential growth. It rises from left to right and becomes increasingly steep. A larger value of $k$ produces faster growth.
• If $k=1$, then for every , so the graph is the horizontal line .

For example, $y=2^x$ and $y=(\frac12)^x$ both pass through . Since , their graphs are reflections of one another in the -axis.

Data for Exponential growth and decay

Sketching from a table of values

To sketch an exponential graph, choose useful $x$-values on both sides of zero and substitute them into the function. For $y=2^x$:

Plot the resulting coordinates, including $(0,1)$, and draw a single smooth curve through them. Continue the shape beyond the plotted points when a sketch is required, showing that the curve approaches the $x$-axis in one direction.

For a decay graph such as $y=(\frac12)^x$, the same process works. Positive powers give $\frac12,\frac14,\frac18,\ldots$, while negative powers give values greater than 1; for example, .

Recognising and interpreting the shape

A rising curve through $(0,1)$ that remains above the $x$-axis is consistent with $y=k^x$ for $k>1$. A falling curve with the same fixed point is consistent with .

Coordinates can reveal the base. On $y=k^x$, the value at $x=1$ is $k^1=k$, so the graph passes through . More generally, dividing the output at one integer -value by the output at the previous integer gives . This constant ratio is the central feature to look for when interpreting an exponential graph.

Core facts

For $y=k^x$, AQA specifies $k>0$.

• Every graph passes through $(0,1)$.