Graph and analyse exponential functions of the form $y = a \cdot b^x + c$, identifying key features (intercepts, asymptote, domain, range) and applying transformations

What this dot point is asking

QCAA wants you to graph exponential functions, identify their key features, and apply standard transformations (vertical and horizontal translations, vertical dilations, reflections) to the parent function $y = b^x$.

The parent function IMATH_1

For base $b > 1$ (for example $y = 2^x$):

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

For $0 < b < 1$ (for example $y = (1/2)^x$):

• Same domain, range, intercept, asymptote.
• Decreasing function. Concave up.

The function value doubles every fixed step in $x$ (for $y = 2^x$, between any two values of $x$ differing by $1$).

Transformations of IMATH_15

y-intercept. Substitute $x = 0$: $y = a b^{-h} + k$.

Horizontal asymptote. Always $y = k$ (the value the function approaches as the exponent goes to negative infinity for $b > 1$, or positive infinity for $b < 1$).

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

Solving graphically and algebraically

To solve $a \cdot b^{x-h} + k = m$, isolate the exponential and either rewrite both sides with the same base or take logarithms (next dot point). Graphically, the solution is the $x$-coordinate where the curve crosses $y = m$.

Worked example

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

• Reflection: $a = -2$ flips the graph vertically.
• Horizontal shift: $h = 2$ shifts the curve $2$ units right.
• Vertical shift: $k = 6$ shifts the curve $6$ units up.
• Horizontal asymptote: $y = 6$, approached from below.
• y-intercept: $y = -2 \cdot 3^{-2} + 6 = -2/9 + 6 = 5.778$.
• As $x \to +\infty$, $3^{x-2} \to +\infty$ and $-2 \cdot 3^{x-2} \to -\infty$, so $y \to -\infty$.
• Decreasing throughout.

Common traps

Confusing the direction of horizontal translation. $y = b^{x-2}$ moves the graph right $2$ units (not left). The graph shifts in the same direction as $h$.

Forgetting to update the asymptote. When you add a constant, the asymptote moves. Failing to update the asymptote in a sketch loses marks.

Mixing the order of transformations. Apply horizontal shift inside the exponent first, then dilations, then vertical shift. Order matters when combining.

Treating the asymptote as a point on the graph. The graph approaches the asymptote but never touches it.

In one sentence

The exponential function $y = a b^{x-h} + k$ has horizontal asymptote $y = k$, y-intercept $a b^{-h} + k$, 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 $0 < b < 1$, and standard transformations apply.