Exponential functions and their graphs (QCE Math Methods Unit 2)

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 $y = b^x$

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 multiplies by the base $b$ over each unit step in $x$, so $y = 2^x$ doubles whenever $x$ increases by $1$. This constant-ratio behaviour is what distinguishes exponential growth from the constant-difference behaviour of a linear function, and it is the reason exponential models eventually outgrow any polynomial.

Transformations of $y = a \cdot b^{x - h} + k$

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 $a < 0$).

Reading the graph's features

The four numbers in $y = a \cdot b^{x - h} + k$ each control a visible feature, which is why a sketch can be built directly from the equation. The base $b$ sets the direction (growing if $b > 1$, decaying if $0 < b < 1$) and the steepness. The constant $k$ sets the horizontal asymptote and the limiting value, $h$ slides the curve sideways, and $a$ scales and possibly flips it. Because the exponential never reaches its asymptote, the range is open on that side: $y > k$ when $a > 0$ and $y < k$ when $a < 0$.

Why the asymptote sits at $y = k$

As the exponent runs to the end of the domain that drives $b^{x - h}$ toward zero (negative infinity for $b > 1$), the term $a \cdot b^{x - h}$ vanishes and only the constant $k$ remains, so the curve flattens toward the line $y = k$. The asymptote is therefore read straight off the constant term, and forgetting to shift it when $k \neq 0$ is the most common sketching error.

Solving graphically and algebraically

To solve $a \cdot b^{x - h} + k = m$, isolate the exponential first, then either rewrite both sides with the same base and equate exponents, or take logarithms (the next dot point). Graphically, the solution is the $x$-coordinate where the curve crosses the horizontal line $y = m$, which exists only when $m$ lies on the correct side of the asymptote.