Why This Matters#
The exponential function models growth that accelerates — or decay that slows down. It describes how populations grow, how diseases spread, how radioactive material decays, and how your money grows with compound interest. Unlike the straight line (constant speed) or parabola (symmetric curve), the exponential is one-directional: it either climbs steeply or drops towards zero.
The Exponential Function: $y = ab^x + q$#
What the Parameters Control#
| Parameter | What it does | How to spot it |
|---|---|---|
| $a$ | Stretch & reflect. $a > 0$: graph is above the asymptote (for growth) or below for decay. $a < 0$: graph is reflected (flipped across the asymptote). | Substitute a known point and solve for $a$. |
| $b$ | Growth or decay. $b > 1$: growth (graph rises steeply to the right). $0 < b < 1$: decay (graph drops towards the asymptote). | Read from the shape: rising = growth, falling = decay. |
| $q$ | Horizontal asymptote. The graph approaches $y = q$ but never touches it. | Read directly from the equation or graph. |
⚠️ The base $b$ must be positive and $b \neq 1$. If $b = 1$, you get $y = a + q$ (a horizontal line, not exponential). If $b < 0$, the function oscillates and is not defined for all real $x$.
Understanding Growth vs Decay#
| Condition | Behaviour | Graph shape |
|---|---|---|
| $b > 1$, $a > 0$ | Growth — rises steeply to the right | Curve climbing away from asymptote |
| $0 < b < 1$, $a > 0$ | Decay — falls towards the asymptote | Curve dropping towards asymptote |
| $b > 1$, $a < 0$ | Reflected growth — drops steeply below asymptote | Mirror of growth, below $y = q$ |
| $0 < b < 1$, $a < 0$ | Reflected decay — rises towards asymptote from below | Mirror of decay, below $y = q$ |
Why does it grow or decay?#
Think of $b^x$ as repeated multiplication:
- If $b = 2$: $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$ — each step doubles. That’s growth.
- If $b = \frac{1}{2}$: $(\frac{1}{2})^1 = 0.5$, $(\frac{1}{2})^2 = 0.25$, $(\frac{1}{2})^3 = 0.125$ — each step halves. That’s decay.
The Asymptote#
Every exponential graph has a horizontal asymptote at $y = q$.
- The graph gets closer and closer to this line but never reaches it.
- Draw it as a dashed line and label it (required in exams).
- There is no vertical asymptote (unlike the hyperbola).
Why $y = q$?#
As $x \to -\infty$ (for growth) or $x \to +\infty$ (for decay), $b^x \to 0$. So $y = a(0) + q = q$. The curve approaches $q$ but the exponential part never actually equals zero.
Key Properties#
| Property | Value |
|---|---|
| Domain | $x \in \mathbb{R}$ (all real numbers) |
| Range | If $a > 0$: $y > q$. If $a < 0$: $y < q$ |
| Horizontal asymptote | $y = q$ |
| $y$-intercept | Let $x = 0$: $y = a(b^0) + q = a + q$ |
| $x$-intercept | Let $y = 0$: solve $0 = ab^x + q$, i.e., $b^x = -\frac{q}{a}$. Only exists if $-\frac{q}{a} > 0$ |
| Shape | Always a smooth curve, never straight |
💡 The $y$-intercept is always at $(0;\, a + q)$ — this is the fastest way to find a key point on the graph.
How to Sketch an Exponential Graph#
- Draw the asymptote $y = q$ as a dashed line and label it.
- Find the $y$-intercept: $(0;\, a + q)$. Plot it.
- Determine growth or decay: Check $b > 1$ (growth) or $0 < b < 1$ (decay).
- Check for reflection: If $a < 0$, the graph is reflected below the asymptote.
- Find the $x$-intercept (if it exists): Solve $ab^x + q = 0$.
- Plot one or two extra points using a table of values.
- Draw a smooth curve approaching the asymptote.
Worked Example 1: Growth#
Sketch $y = 2^x + 1$
Here $a = 1$, $b = 2$, $q = 1$.
Step 1: Asymptote: $y = 1$ (dashed line)
Step 2: $y$-intercept: $y = 1 + 1 = 2$. Plot $(0;\, 2)$.
Step 3: $b = 2 > 1$ and $a = 1 > 0$ → growth curve above asymptote.
Step 4: $x$-intercept: $0 = 2^x + 1 \Rightarrow 2^x = -1$ — impossible! No $x$-intercept.
Step 5: Extra points:
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|
| $y$ | $1.25$ | $1.5$ | $2$ | $3$ | $5$ | $9$ |
The graph starts near $y = 1$ on the left and climbs steeply to the right.
Worked Example 2: Decay#
Sketch $y = 3 \cdot \left(\frac{1}{2}\right)^x - 1$
Here $a = 3$, $b = \frac{1}{2}$, $q = -1$.
Step 1: Asymptote: $y = -1$
Step 2: $y$-intercept: $y = 3(1) - 1 = 2$. Plot $(0;\, 2)$.
Step 3: $0 < b < 1$ and $a > 0$ → decay curve above asymptote.
Step 4: $x$-intercept: $0 = 3(\frac{1}{2})^x - 1 \Rightarrow (\frac{1}{2})^x = \frac{1}{3}$
Using trial: $(\frac{1}{2})^1 = 0.5$, $(\frac{1}{2})^{1.58} \approx 0.33$. So $x \approx 1.58$.
Step 5: Extra points:
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|
| $y$ | $11$ | $5$ | $2$ | $0.5$ | $-0.25$ | $-0.625$ |
The graph starts high on the left and falls towards $y = -1$.
Worked Example 3: Finding the Equation#
A graph has asymptote $y = 2$, passes through $(0;\, 5)$ and $(1;\, 8)$. Find the equation.
Since $q = 2$: $y = ab^x + 2$
Using $(0;\, 5)$: $5 = a \cdot b^0 + 2 = a + 2 \Rightarrow a = 3$
Using $(1;\, 8)$: $8 = 3b + 2 \Rightarrow 3b = 6 \Rightarrow b = 2$
Equation: $y = 3 \cdot 2^x + 2$
Worked Example 4: Reflected Exponential#
Sketch $y = -2^x + 4$
Here $a = -1$, $b = 2$, $q = 4$.
Step 1: Asymptote: $y = 4$
Step 2: $y$-intercept: $y = -1 + 4 = 3$. Plot $(0;\, 3)$.
Step 3: $a < 0$ → reflected below the asymptote. Graph is BELOW $y = 4$.
Step 4: $x$-intercept: $0 = -2^x + 4 \Rightarrow 2^x = 4 \Rightarrow x = 2$. Plot $(2;\, 0)$.
Range: $y < 4$ (everything below the asymptote).
Reading Information from Exponential Graphs#
| Question | Method |
|---|---|
| Find the asymptote | Read the horizontal line the graph approaches |
| Find $q$ | $q$ = the asymptote value |
| Find $a$ | Use the y-intercept: $a = y\text{-int} - q$ |
| Find $b$ | Substitute another point into $y = ab^x + q$ and solve for $b$ |
| Domain | Always $x \in \mathbb{R}$ |
| Range | If $a > 0$: $y > q$. If $a < 0$: $y < q$ |
| $f(x) > 0$ | Where the graph is above the $x$-axis |
The Connection to Compound Interest#
The compound interest formula $A = P(1 + i)^n$ IS an exponential function:
- $P$ plays the role of $a$ (the starting amount)
- $(1 + i)$ plays the role of $b$ (the growth factor)
- $n$ plays the role of $x$ (time)
This is why compound interest grows faster and faster — it’s exponential growth in action.
🚨 Common Mistakes#
- Forgetting the asymptote: EVERY exponential graph has a horizontal asymptote at $y = q$. You MUST draw it as a dashed line and label it.
- Confusing growth and decay: $b > 1$ = growth, $0 < b < 1$ = decay. If $b = \frac{1}{3}$, it’s decay (the base is between 0 and 1).
- Drawing through the asymptote: The graph approaches but NEVER touches or crosses the asymptote (unless the reflection makes it cross the $x$-axis, which is different).
- $x$-intercept doesn’t always exist: If $a > 0$ and $q > 0$ (both positive), then $ab^x + q > 0$ always — no $x$-intercept.
- $y$-intercept is NOT $q$: The $y$-intercept is $a + q$ (substitute $x = 0$). The asymptote is $q$.
- Drawing with a ruler: An exponential is a smooth CURVE that gets steeper (growth) or flatter (decay). Never use a ruler.
💡 Pro Tip: The “One Step” Check#
For any exponential $y = ab^x + q$:
- If you move one unit right from any point, the $y$-value gets multiplied by $b$ (relative to the asymptote).
This means: from the $y$-intercept $(0; a+q)$, going right 1 gives $(1; ab + q)$. The distance from the asymptote goes from $a$ to $ab$. This multiplicative pattern is what makes exponentials unique.
🔗 Related Grade 10 topics:
- Sketching Graphs (Linear, Quadratic, Hyperbola) — the other three functions you must sketch
- Simple & Compound Interest — compound interest IS an exponential function
- Exponent Laws — understanding $b^x$ requires solid exponent skills
📌 Where this leads in Grade 11: The Exponential Function — adds horizontal shift $p$: $y = ab^{x-p} + q$
📌 Where this leads in Grade 12: Exponential Function & Inverse — leads into logarithms