Exponential graphs

The idea

$y = e^{kx}$ passes through $(0, 1)$ and has the $x$ -axis as an asymptote. If $k > 0$ it grows; if $k < 0$ it decays. Adding a constant moves the asymptote: $y = Ae^{kx} + c$ has asymptote $y = c$ . The gradient of $e^{kx}$ is $ke^{kx}$ .

Put $x = 0$ for the $y$ -intercept: $A + c$ .
The asymptote is $y = c$ .
The sign of $k$ tells you growth or decay.

Watch out An added constant moves the asymptote off the $x$ -axis.

See it worked

Worked example

Describe the graph of $y = 3e^{-x} + 2$ .

$y$ -intercept: $3 + 2 = 5$ .
As $x$ grows, $3e^{-x} \to 0$ , so the asymptote is $y = 2$ .
$k = -1 < 0$ , so the curve decreases towards $y = 2$ .

Answer Through $(0, 5)$ , decreasing towards the asymptote $y = 2$

Your turn

Question 1Non-calculator3 marks

Find where $y = 4e^{2x} - 1$ crosses the $y$ -axis.

Question 2Non-calculator3 marks

Write down the equation of the asymptote of $y = 5 - 2e^{-x}$ .

Question 3Non-calculator3 marks

Find the gradient of $y = e^{3x}$ at the point where $x = 0$ .

In past papers

13 official questions in 13 papers since 2018. The latest:

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