Graphs with asymptotes: exponentials and hyperbolas

Sketch exponential graphs and rectangular hyperbolas, find their asymptotes, and use limit notation to describe what happens at the edges.

What is an asymptote?

An is a line the graph approaches more and more closely but never reaches. Draw asymptotes as dashed lines and label their equations.

We describe this approach with limit notation. "As $x \to \infty$, $y \to 0$" means: as $x$ gets larger and larger, $y$ gets closer and closer to $0$.

Exponential functions

An exponential function has the variable in the index: $y = a^x$ with $a > 0$ and $a \ne 1$.

It always passes through $(0, 1)$, because $a^0 = 1$.
It is always positive, so its range is $y > 0$.
The $x$-axis ($y = 0$) is a horizontal asymptote.
If $a > 1$ it increases (growth). If $0 < a < 1$ it decreases (decay).

Exponential growth and decay

Try this

Which point stays fixed as you change $a$?
Compare $a = 2$ with $a = 0.5$. How are the two graphs related?

The rectangular hyperbola

The graph of $y = \dfrac{k}{x}$ is a rectangular hyperbola. It has two separate branches, and both axes are asymptotes:

Vertical asymptote $x = 0$, because we cannot divide by zero.
Horizontal asymptote $y = 0$, because $\dfrac{k}{x} \to 0$ as $x \to \pm\infty$.

If $k > 0$ the branches are in the 1st and 3rd quadrants; if $k < 0$ they are in the 2nd and 4th.

Shift a hyperbola

The dashed pink lines are the asymptotes. They move with the graph.

Try this

Find where this hyperbola crosses the $y$-axis by hovering. Check it by substituting $x = 0$.
Set $k$ negative. Which quadrants (relative to the asymptotes) do the branches move to?

Example 1●●

Sketch $y = \dfrac{3}{x - 2} + 1$, showing asymptotes and intercepts.

Vertical asymptote: $x = 2$.

The denominator is zero when x = 2.

Horizontal asymptote: $y = 1$.

As x → ±∞, the fraction → 0, so y → 1.

$y$-intercept: $x = 0$ gives $y = \tfrac{3}{-2} + 1 = -\tfrac12$.

$x$-intercept: $\dfrac{3}{x - 2} = -1$, so $x - 2 = -3$ and $x = -1$.

Set y = 0 and solve.

Sketch the dashed asymptotes, mark $(0, -\tfrac12)$ and $(-1, 0)$, then draw the two branches.

k = 3 > 0, so the branches sit top-right and bottom-left of where the asymptotes cross, at (2, 1).

Example 1 sketch

Example 2●●

Sketch $y = 2^x - 4$ and state its range.

Horizontal asymptote $y = -4$.

Subtracting 4 shifts y = 2ˣ down 4 units, so the asymptote moves from y = 0 to y = −4.

$y$-intercept: $2^0 - 4 = -3$.

$x$-intercept: $2^x = 4$, so $x = 2$.

Range: $y > -4$.

Flashcards

Asymptote

A line the graph approaches but never reaches. Draw it dashed and label it.

y = aˣ passes through

$(0, 1)$, with asymptote $y = 0$.

Asymptotes of y = k/(x − h) + c

$x = h$ and $y = c$

x → ∞

"$x$ gets larger without bound."

Range of y = 2ˣ − 4

$y > -4$

Practice

The horizontal asymptote of $y = 3^x + 2$ is:
$y = \dfrac{-2}{x}$ has branches in which quadrants?
$y = \dfrac{1}{x + 3}$ has a vertical asymptote $x = \ ?$
Every graph $y = a^x$ (with $a > 0$, $a \ne 1$) passes through:
The range of $y = 2^{-x}$ is:
Find the $y$-intercept of $y = 3^x - 5$.
As $x \to \infty$, $\dfrac{4}{x} \to$
Find the $x$-intercept of $y = \dfrac{2}{x + 1} - 1$.