How Do You Graph an Exponential Function?

To graph an exponential function such as $y=a^x$, determine whether it represents growth or decay, plot several key points, and draw a smooth curve approaching its horizontal asymptote. For $a>0$ and $a\neq1$, the graph always passes through $(0,1)$.

This is IB Mathematics: Analysis and Approaches SL, sub-topic SL 2.9: Exponential and logarithmic functions, and it can be tested on Paper 1 or Paper 2.

An exponential function has the variable in the exponent. For the basic function

$$y=a^x,$$

substitute convenient integer values such as $x=-2,-1,0,1,2$. For example, for $y=2^x$:

Plot these points and join them with a smooth curve. The graph approaches but never reaches the horizontal asymptote $y=0$.

The domain is all real numbers, while the range is $y>0$. The function $y=e^x$ has the same basic growth shape because $e>1$.

Why these features occur: $a^0=1$, so the $y$-intercept is $(0,1)$. Negative exponents give reciprocals because $a^{-x}=\frac{1}{a^x}$, explaining why outputs approach zero on one side. However, is positive for every real , so it never equals zero. This establishes both the range and asymptote.

For $y=p\,a^{x-h}+k$, the asymptote is $y=k$; $h$ shifts horizontally, while gives a vertical stretch and reflects the graph when .

For a transformed function such as $y=3(2^x)+4$, multiply every output by $3$, then translate the graph upward by $4$. Its horizontal asymptote becomes $y=4$, and its range becomes .

A common misconception is that an exponential graph touches its horizontal asymptote. It does not: the function approaches the asymptote indefinitely.

Exam technique: On a sketch, label the axes, the $y$-intercept, and the horizontal asymptote. The command term sketch requires the correct general shape and relevant features, not a perfectly scaled graph. On Paper 2, technology may be used, but key features should still be identified clearly.