Exponential Graphs y = ab^(x + p) + q

Q: How do I find the asymptote of an exponential graph?

It is $y = q$, the constant added at the end of the equation.

Q: What is the difference between b > 1 and 0 < b < 1?

With $b > 1$ the basic graph grows; with $0 < b < 1$ it decays. A negative $a$ reverses this.

Exponential graphs show quantities that grow or decay by a constant factor, like populations or investments. In Grade 11 you sketch $y = a b^{x + p} + q$ , find its asymptote and intercepts, and work out the equation from given points.

The effect of a, b, p and q

For $y = a b^{x + p} + q$ , with $b > 0$ and $b \neq = 1$ :

$b > 1$ : the graph of $b^{x}$ increases. $0 < b < 1$ : it decreases.
$a < 0$ : reflects the graph in the asymptote, so increasing becomes decreasing.
$p$ : shifts the graph horizontally ( $p$ units left if $p > 0$ ).
$q$ : shifts the graph and the asymptote vertically.

Example. $y = 3 \cdot 2^{x} - 6$ increases, with asymptote $y = - 6$ .

Asymptote, intercepts and sketching

Asymptote: $y = q$
Range: $y > q$ if $a > 0$ , or $y < q$ if $a < 0$
$y$ -intercept: let $x = 0$
$x$ -intercept: let $y = 0$ ; write both sides with the same base

Example. $y = 3 \cdot 2^{x} - 6$

$y$ -intercept: $3 - 6 = - 3$
$x$ -intercept: $3 \cdot 2^{x} = 6 \Rightarrow 2^{x} = 2 \Rightarrow x = 1$

Not every exponential graph has an $x$ -intercept. If $a > 0$ and $q \geq 0$ , the graph stays above the $x$ -axis.

To sketch: draw the asymptote as a dotted line, plot the intercepts and one more point, and draw a smooth curve approaching the asymptote.

Finding the equation from a graph

Read $q$ from the asymptote.
Use the given points to set up equations in $a$ and $b$ .

Example. $y = a \cdot b^{x} + q$ has asymptote $y = - 1$ and passes through $(0; 1)$ and $(1; 3)$ .

$q = - 1$
$(0; 1)$ : $a - 1 = 1$ , so $a = 2$
$(1; 3)$ : $2 b - 1 = 3$ , so $b = 2$

$y = 2 \cdot 2^{x} - 1 = 2^{x + 1} - 1$

Key terms

Exponential function: A function with the variable in the exponent.
Base: The number $b$ in $b^{x}$ ; it must be positive and not 1.
Horizontal asymptote: The line $y = q$ that the graph approaches.

Common mistakes

Thinking every exponential graph has an $x$ -intercept.
Using the $y$ -intercept as $a$ when $q \neq = 0$ .
Drawing the graph crossing its asymptote.
Forgetting that $b^{0} = 1$ when finding the $y$ -intercept.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

For $y = 3 \cdot 2^{x} - 6$ , write down:

(a) the equation of the asymptote (1)

(b) the $y$ -intercept (1)

Worked solution

Hidden - try the question first

Solution

(a) $y = - 6$
(b) $3 (1) - 6 = - 3$
(c) $y > - 6$

Question 2

5 marks total

routine

For $y = 3 \cdot 2^{x} - 6$ :

(a) Determine the $x$ -intercept. (3)

(b) Sketch the graph. (2)

Worked solution

Hidden - try the question first

Solution

(a) $y = - 6$
$2^{x} = 2$
$x = 1$
(b) Asymptote $y = - 6$ drawn dotted; intercepts $(0; - 3)$ and $(1; 0)$
Increasing curve approaching the asymptote on the left

Question 3

6 marks total

complex

Given $g (x) = - (\frac{1}{2})^{x - 1} + 4$ .

(a) Write down the asymptote. (1)

(b) Determine the intercepts with the axes. (4)

Worked solution

Hidden - try the question first

Solution

(a) $y = - 6$
(b) $y$ -intercept: $- (\frac{1}{2})^{- 1} + 4 = - 2 + 4 = 2$
$x$ -intercept: $(\frac{1}{2})^{x - 1} = 4 = (\frac{1}{2})^{- 2}$
$x - 1 = - 2$
$x = - 1$
(c) Increasing ( $(\frac{1}{2})^{x}$ decreases, and the negative $a$ reflects it)

Question 4

8 marks total

problem-solving

The graph of $y = a \cdot b^{x} + q$ has asymptote $y = - 1$ and passes through $(0; 1)$ and $(1; 3)$ .

(a) Determine $a$ , $b$ and $q$ . (4)

(b) For which values of $x$ is $y > 15$ ? (2)

Worked solution

Hidden - try the question first

Solution

(a) $y = - 6$
$1 = a - 1 \Rightarrow a = 2$
$3 = 2 b - 1$
$b = 2$ , so $y = 2 \cdot 2^{x} - 1$
(b) $2^{x + 1} - 1 > 15 \Rightarrow 2^{x + 1} > 16 = 2^{4}$
$x > 3$
(c) Replace $x$ with $- x$ :
$y = 2 \cdot 2^{- x} - 1$

FAQs

How do I find the asymptote of an exponential graph?

It is $y = q$ , the constant added at the end of the equation.

Do I need logarithms for exponential graphs in Grade 11?

No. In Grade 11 you write both sides with the same base. Logarithms come in Grade 12.

What is the difference between b > 1 and 0 < b < 1?

With $b > 1$ the basic graph grows; with $0 < b < 1$ it decays. A negative $a$ reverses this.

The effect of a, b, p and q

Asymptote, intercepts and sketching

Finding the equation from a graph

Key terms

Common mistakes

Practice questions

FAQs

Need a tutor for this subject?