Read the multiplier.

In y = $r^{x}$ with r > 0 and r ≠ 1, the y-intercept is 1 and the horizontal asymptote is y = 0. A base above one produces growth, while a base between zero and one produces decay.

The transformed rule y = r^(x − h) + k has horizontal asymptote y = k. Exponential models use equal time steps and a fixed multiplicative rate. Solving for time usually needs logarithms, introduced next.

INTERACTIVE MODEL

Try it. Watch it change.

y =

3.2^{x}

+ 0

Base control (r shown above)1 Vertical shift k0Input x1

f(1) = 3.2 · asymptote y = 0

The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.

Explore: Change the base and vertical shift. Identify the horizontal asymptote each time.

Exponential model

N(t)=N_0r^t

WORKED EXAMPLE

Solve 3^(x − 1) = 27.

Write 27 = $3^{3}$ .
Equate exponents: x − 1 = 3, so x = 4.

Assumed knowledge

Indices and transformations.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Transform exponential functions and identify asymptotes.
Solve growth and decay models.

QCAA Mathematical Methods 2025 v1.3 · p. 20 ↗

READY TO TRY IT?

Read the multiplier.

Solve 3^(x − 1) = 27.

Put the idea to work.