Transformations of exponential functions

Powers of e on the calculator, the graphs of y = e^x and y = e^(−x), and reflecting, translating and dilating exponential graphs.

Powers of e

Use the e^x key (often SHIFT + ln): $e^2 \approx 7.389$, $e^{-1} \approx 0.368$, $e^{0.5} \approx 1.649$.

The graphs of e^x and e^(−x)

$y = e^{-x}$ is the reflection of $y = e^x$ in the $y$-axis. It is always decreasing, passes through $(0, 1)$ with gradient $-1$, and has asymptote $y = 0$ as $x \to \infty$. It models decay.

Transforming y = eˣ

Faint pink: y = eˣ. Orange: y = c·e^(±(x − h)) + k. Dashed: the asymptote.

Try this

Set the sign of $x$ to $-1$. What changes?
Make $c$ negative. Which side of the asymptote is the curve now?

Example 1●●

Sketch $y = 2 - e^{-x}$.

Start with $e^{-x}$ (decreasing), reflect in the $x$-axis to get $-e^{-x}$ (increasing, below the axis), then shift up $2$.

Asymptote $y = 2$ as $x \to \infty$; $y$-intercept $2 - 1 = 1$; $x$-intercept where $e^{-x} = 2$, $x = -\ln 2 \approx -0.69$.

Flashcards

$y = e^{-x}$

Reflection of $e^x$ in the $y$-axis; decay.

$y = e^x + k$

Asymptote $y = k$

$e^{-1}$

$\approx 0.368$

Practice

$y = e^{-x}$ is the reflection of $y = e^x$ in:
$y = e^x - 2$ has horizontal asymptote $y = k$. Find $k$.
Find the $y$-intercept of $y = 3e^x + 1$.
$y = e^{x - 2}$ is $y = e^x$ shifted:
Find $e^{-1}$, correct to 3 decimal places.
The range of $y = -e^x$ is:
Find the $y$-intercept of $y = e^{x + 1}$, correct to 3 decimal places.
$y = 2 - e^{-x}$ is:
Find the $x$-intercept of $y = e^x - 1$.