Transformations and asymptotes of exponential graphs | Year 11

01 / LEARN

The idea, step by step

Move the graph while tracking its features

For $y = a b^{x - h} + k$ , with $a \neq 0$ , $b > 0$ and $b \neq 1$ , the number k is the horizontal asymptote. The quantity $b^{x - h}$ stays positive; a changes the vertical scale and may reflect the curve.

The expression x-h shifts the parent curve right by h. Identify the transformation from the brackets rather than treating h as the y-intercept. Substitute x=0 separately to find that intercept.

y = a b^{x - h} + k

Explain domain and range from positivity

An exponential expression with a valid positive base is defined for every real x. If a is positive, its outputs are above k; if a is negative, they are below k. The asymptote is approached, not reached.

Use one or two exact reference points and the asymptote to sketch. A graphing tool can test the sketch, but a screen window is not a reason to truncate the mathematical domain or range.

A common trap

Including the horizontal asymptote in the range.

Instead: The exponential term is never zero, so the shifted curve never attains its asymptote.

02 / WORKED EXAMPLES

See the thinking

Example 1

Shift and stretch

Describe $g (x) = 3 (2^{x - 1}) - 4$ and find its y-intercept.

From $2^{x}$ : shift right 1, stretch vertically by 3 and move down 4.
$g (0) = 3 (2^{- 1}) - 4 = - 2.5$ . The asymptote is y=-4.
The domain is all real numbers and range is $y > - 4$ .

Intercept (0,-2.5); asymptote y=-4; range above -4.

Example 2

Reflect an exponential

Find the range and x-intercept of $h (x) = 5 - 2^{x}$ .

Since $2^{x} > 0$ , subtracting it gives $h (x) < 5$ .
For an intercept, $2^{x} = 5$ , so $x = log (5) / log (2) \approx 2.322$ .

Range

(- \infty, 5)

; x-intercept approximately (2.322,0).

Check your understanding

What is the asymptote of $y = 4 {(0.5)}^{x + 2} + 7$ ?

Show answer

$y = 7$

The added constant gives the horizontal asymptote.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the domain and range of $y = 2^{x} + 3$ .
2 marks
Find the y-intercept of $y = 4 (3^{x - 1}) - 2$ .
2 marks
Describe the transformations from $y = 3^{x}$ to $y = - 2 (3^{x + 1}) + 4$ .
2 marks
Find the asymptote and range of $y = 6 - 4 {(0.8)}^{x}$ .
2 marks
Find the x-intercept of $y = 2^{x + 1} - 8$ .
2 marks
Explain why $y = 2^{x} + 1$ is not the same graph as $y = 2^{x + 1}$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the domain and range of $y = 2^{x} + 3$ .
2 marks
$2^{x}$ is defined and positive for all real x.
Domain $ℝ$ ; range $y > 3$ .
Find the y-intercept of $y = 4 (3^{x - 1}) - 2$ .
2 marks
At x=0, $4 / 3 - 2 = - 2 / 3$ .
$(0, - 2 / 3)$
Describe the transformations from $y = 3^{x}$ to $y = - 2 (3^{x + 1}) + 4$ .
2 marks
The inner x+1 shifts left; the outer -2 stretches and reflects.
Left 1; vertical stretch factor 2; reflection in the x-axis; up 4.
Find the asymptote and range of $y = 6 - 4 {(0.8)}^{x}$ .
2 marks
Four times the exponential is positive, so subtracting it stays below 6.
Asymptote y=6; range $y < 6$ .
Find the x-intercept of $y = 2^{x + 1} - 8$ .
2 marks
$2^{x + 1} = 8 = 2^{3}$ , so x+1=3 and x=2.
(2,0)
Explain why $y = 2^{x} + 1$ is not the same graph as $y = 2^{x + 1}$ .
3 marks
The first adds 1 to every output; the second multiplies $2^{x}$ by 2.
They also have different values at x=1: 3 and 4.
The first has asymptote y=1; the second has asymptote y=0.

Checking saved progress…

Continue learning

Fitting/interpreting exponential models

Next lesson

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: exponential and logarithmic functions, growth models and numerical/algebraic prerequisites.

Official curriculum source ↗

Back to Exponentials and logarithms