Asymptotes and shifts of exponential and logarithmic graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = a \cdot 2^x + k$ .

The graph of y = a·2^x + k, which changes as you move the sliders for a and k.

What does $k$ change?

It moves the graph and its horizontal asymptote up or down
It only moves the point where the graph crosses the y-axis
It makes the graph steeper

Key idea

Shifts move the asymptotes with the graph.

y = a \cdot b^{x} + k

has the horizontal asymptote

y = k

.

y = \log_b(x - h) + k

has the vertical asymptote

x = h

and domain

x > h

A vertical stretch by $a$ does not move a horizontal asymptote.

Worked example

Describe $y = 3 \cdot 2^x + 4$ .

Horizontal asymptote:

y = 4

As x → −∞, 3 · 2^x → 0.

y

-intercept:

3 + 4 = 7

At x = 0, 2⁰ = 1.

Your turn, guided

Your turn: $y = \log_3(x - 2) + 5$ . Fill in the missing line.

Domain:

x > 2

The argument x − 2 must be positive.

Vertical asymptote:

x = 2

The asymptote is where the argument equals 0.

Practice ★

The graph of $y = 3\cdot 4^x + 7$ has a horizontal asymptote $y = c$ . What is $c$ ?
The graph of $y = 8\cdot 5^x + 5$ has a horizontal asymptote $y = c$ . What is $c$ ?
The graph of $y = 2\cdot 3^x + 9$ has a horizontal asymptote $y = c$ . What is $c$ ?
The graph of $y = 6\cdot 5^x + 2$ has a horizontal asymptote $y = c$ . What is $c$ ?
The graph of $y = 7\cdot 2^x + 2$ has a horizontal asymptote $y = c$ . What is $c$ ?

Watch out!

Practice ★★

Draw the graph of $y = 1 \cdot 2^x + 3$ : place two points and the asymptote.
Draw the graph of $y = 1 \cdot 2^x + 1$ : place two points and the asymptote.

The graph of $y = \log_{ 4 }(x - 6) + 1$ has a vertical asymptote $x = c$ . What is $c$ ?
The graph of $y = \log_{ 2 }(x - 4) + 8$ has a vertical asymptote $x = c$ . What is $c$ ?
The graph of $y = \log_{ 2 }(x - 9) + 3$ has a vertical asymptote $x = c$ . What is $c$ ?
The graph of $y = \log_{ 4 }(x - 9) + 5$ has a vertical asymptote $x = c$ . What is $c$ ?
The graph of $y = \log_{ 4 }(x - 5) + 3$ has a vertical asymptote $x = c$ . What is $c$ ?

Exit ticket

Three questions to check you have got it.

The graph of $y = \log_{ 4 }(x - 6) + 1$ has a vertical asymptote $x = c$ . What is $c$ ?
The graph of $y = \log_{ 2 }(x - 4) + 8$ has a vertical asymptote $x = c$ . What is $c$ ?
The graph of $y = \log_{ 2 }(x - 9) + 3$ has a vertical asymptote $x = c$ . What is $c$ ?

Summary card

Asymptotes after shifts

$y = a \cdot b^{x} + k$ : horizontal asymptote $y = k$ .
$y = \log_b(x - h) + k$ : vertical asymptote $x = h$ , domain $x > h$ .

About this lesson