Sketch and analyse exponential functions of the form $y = a \cdot b^{x - h} + k$, identifying key features (intercepts, asymptote, domain, range) and applying transformations

What this dot point is asking

VCAA wants you to graph exponential functions, identify their key features (intercepts, asymptote, domain, range), and apply standard transformations.

Parent function IMATH_0

For $b > 1$:

• Domain: all real $x$. Range: $y > 0$.
• y-intercept: $(0, 1)$.
• Horizontal asymptote: $y = 0$ as $x \to -\infty$.
• Increasing, concave up.

For $0 < b < 1$:

• Same domain, range, intercept and asymptote.
• Decreasing, concave up.

Transformed form IMATH_8

• IMATH_9 vertical dilation by factor $|a|$ (reflection in $x$-axis if $a < 0$).
• IMATH_13 horizontal shift right by $h$ units.
• IMATH_15 vertical shift up by $k$ units; new asymptote $y = k$.

Key features

y-intercept: $f(0) = a b^{-h} + k$.

Horizontal asymptote: $y = k$ (always).

Range: $y > k$ (if $a > 0$) or $y < k$ (if $a < 0$).

Solving exponential equations

If both sides can be rewritten in the same base, equate exponents. Otherwise take logs (next dot point).

Worked example

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

Parameters: $a = -2$, $h = 2$, $k = 6$.

• Asymptote: $y = 6$, approached from below.
• y-intercept: $-2 \cdot 3^{-2} + 6 = -2/9 + 6 = 5.78$.
• As $x \to +\infty$, $3^{x-2} \to \infty$, so $-2 \cdot 3^{x-2} \to -\infty$ and the graph descends.
• Decreasing throughout.

Common traps

Direction of horizontal translation. $y = b^{x-2}$ shifts right by $2$, not left.

Forgetting to update the asymptote. When adding $k$, the asymptote moves with it.

Mixing the order of transformations. Apply horizontal shift inside the exponent, then dilation, then vertical shift.

In one sentence

The exponential function $y = ab^{x-h} + k$ has horizontal asymptote $y = k$, y-intercept $ab^{-h} + k$, domain all real $x$, and range $y > k$ if $a > 0$ (or $y < k$ if $a < 0$); the parent $y = b^x$ is increasing for $b > 1$ and decreasing for $0 < b < 1$.