Sketching Exponential Functions Revision Notes | GCSE AQA Higher Further Maths

1. What Makes Exponential Functions Different?

You have already worked with linear graphs (straight lines) and quadratic graphs (parabolas). Exponential functions behave in a completely different way — instead of adding a fixed amount each time, they multiply by a fixed amount. That single difference produces a shape unlike anything else on the GCSE specification.

Think about a rumour spreading through a school. On day one, one person knows. On day two, two people know. On day three, four people know. The number keeps doubling — that is exponential growth in action. The graph of this process curves upward with increasing steepness, rather than climbing at a steady rate.

The key distinction from a quadratic like $y = x^2$ is this: in a quadratic, $x$ is the base and $2$ is the fixed power. In an exponential like $y = 2^x$, the base $2$ is fixed and $x$ is the power. That swap changes everything about the graph's behaviour.

2. The Standard Forms: $y = ab^x$ and $y = ab^{-x}$

At GCSE, exponential graphs take one of two forms. The first is $y = ab^x$, where $a$ and $b$ are rational numbers, $b > 0$, and $b eq 1$. The second is $y = ab^{-x}$, which introduces a negative exponent. Understanding what each part of the equation controls is the foundation for sketching these graphs accurately.

The value of $a$ controls where the graph crosses the $y$-axis. Since $b^0 = 1$ for any valid base $b$, substituting $x = 0$ gives $y = a \times 1 = a$. So the $y$-intercept is always the point $(0, a)$.

The value of $b$ controls the rate of change. When $b > 1$, each step to the right multiplies $y$ by $b$, so the graph climbs. When $0 < b < 1$, each step to the right multiplies $y$ by a fraction less than one, so the graph falls. A negative exponent in $y = ab^{-x}$ has the same effect as replacing $b$ with $\frac{1}{b}$, which flips the direction of the curve.

3. Sketching Exponential Graphs: Key Features

When the exam asks you to sketch a graph, it wants the correct shape with key features labelled — not a precise point-by-point plot. For exponential graphs, those key features are: the $y$-intercept, the horizontal asymptote, and the overall direction of the curve.

• $y$-intercept: always at $(0, a)$ — substitute $x = 0$ to confirm.
• Horizontal asymptote: the curve approaches $y = 0$ (the $x$-axis) but never crosses it.
• Direction: rises steeply to the right for growth ($b > 1$); falls and flattens to the right for decay.
• No $x$-intercept: the curve never touches the $x$-axis when $a > 0$. (If $a < 0$, the curve lies below the $x$-axis but still has asymptote $y = 0$.)
• The curve passes through $(1, ab)$ for $y = ab^x$ — a useful second point to mark.

The asymptote is the feature students most often forget to mark. Draw a dashed line along $y = 0$ and label it — examiners award marks for this. The curve must visibly approach but not touch or cross that line.

Worked example: Sketching $y = 2 \times 3^x$

Sketch the graph of $y = 2 \times 3^x$, labelling the $y$-intercept and the asymptote.

1. $a = 2$, $b = 3$ — These two values control the intercept and the rate of growth. Since $b = 3 > 1$, this is an exponential growth curve. [M1 — identifying key parameters]
2. $x = 0$: $y = 2 \times 3^0 = 2 \times 1 = 2$, so the point is $(0, 2)$ — Any base raised to the power zero equals 1. Mark $(0, 2)$ on the $y$-axis. [A1 — correct intercept]
3. $x = 1$: $y = 2 \times 3^1 = 2 \times 3 = 6$, so the point is $(1, 6)$ — A second point confirms the steepness of the curve and helps you draw the correct shape.
4. $x = -1$: $y = 2 \times 3^{-1} = 2 \times \frac{1}{3} = \frac{2}{3} \approx 0.67$ — As $x$ decreases, $y$ gets smaller and closer to zero — confirming the asymptote at $y = 0$.
5. Draw a dashed line along $y = 0$ and label it 'asymptote: $y = 0$' — The curve approaches this line from above but never touches it. Missing this label costs marks. [A1 — asymptote labelled]
6. Draw a smooth curve through $(0, 2)$ and $(1, 6)$, rising steeply to the right and flattening towards $y = 0$ to the left. — The curve should be smooth — no sharp corners or straight-line segments. [A1 — correct shape]

Answer: A smooth exponential growth curve crossing the $y$-axis at $(0, 2)$, passing through $(1, 6)$, with a horizontal asymptote at $y = 0$.

4. Exponential Decay: The Graph of $y = ab^{-x}$

Now that you can sketch a growth curve, the decay form follows the same logic — but the curve runs in the opposite direction. In $y = ab^{-x}$, the negative exponent means that as $x$ increases, the power becomes more negative, so $b^{-x}$ shrinks towards zero. The graph is high on the left, falls steeply, and flattens out towards $y = 0$ on the right.

Notice that $y = ab^{-x}$ is equivalent to $y = a \left(\frac{1}{b}\right)^x$. So a decay curve with base $b$ and a negative exponent behaves identically to a growth curve with base $\frac{1}{b}$, where $0 < \frac{1}{b} < 1$. Both descriptions produce the same shape.

Worked example: Sketching $y = 4 \times 2^{-x}$

Sketch the graph of $y = 4 \times 2^{-x}$, labelling the $y$-intercept and the asymptote.

1. $a = 4$, $b = 2$, negative exponent — Since $b = 2 > 1$ and the exponent is negative, this is an exponential decay curve. [M1]
2. $x = 0$: $y = 4 \times 2^{0} = 4 \times 1 = 4$, so the point is $(0, 4)$ — $2^{-0} = 2^0 = 1$. The $y$-intercept is $(0, a) = (0, 4)$. [A1]
3. $x = 1$: $y = 4 \times 2^{-1} = 4 \times \frac{1}{2} = 2$, so the point is $(1, 2)$ — The value halves each time $x$ increases by 1 — confirming decay behaviour.
4. $x = 3$: $y = 4 \times 2^{-3} = 4 \times \frac{1}{8} = \frac{1}{2} = 0.5$ — As $x$ grows, $y$ approaches zero — confirming the asymptote at $y = 0$.
5. Draw a dashed line along $y = 0$ and label it 'asymptote: $y = 0$' — The curve approaches $y = 0$ from above as $x \to +\infty$. [A1 — asymptote]
6. Draw a smooth curve through $(0, 4)$ and $(1, 2)$, falling steeply from the left and flattening towards $y = 0$ on the right. — The curve falls from upper-left to lower-right, always staying above the $x$-axis. [A1 — correct shape]

Answer: A smooth exponential decay curve crossing the $y$-axis at $(0, 4)$, passing through $(1, 2)$, with a horizontal asymptote at $y = 0$.

5. Interpreting Exponential Graphs

Reading and interpreting an exponential graph is a separate skill from sketching one. In an exam, you might be given a graph and asked to read off values, describe the behaviour, or identify the equation. Each of these tasks uses the key features you have already practised.

To read a value from the graph, locate the $x$-value on the horizontal axis, move vertically to the curve, then read across to the $y$-axis. To identify the equation, find the $y$-intercept to determine $a$, then use a second point to work out $b$.

Worked example: Identifying the Equation from a Graph

A graph passes through $(0, 3)$ and $(1, 6)$. The graph has the form $y = ab^x$. Find the values of $a$ and $b$.

1. $(0, 3)$ is on the curve, so $3 = a \times b^0 = a \times 1 = a$ — $b^0 = 1$ always, so the $y$-intercept directly gives $a = 3$. [M1]
2. $(1, 6)$ is on the curve: $6 = 3 \times b^1 = 3b$ — Now we have a simple equation to solve for $b$. [M1]
3. $b = \frac{6}{3} = 2$ — Dividing both sides by 3 gives $b = 2$. [A1]
4. $y = 3 \times 2^x$ — Always write the full equation with both values clearly stated. [A1]

Answer: $y = 3 \times 2^x$

The same method works for decay graphs. If the curve passes through $(0, a)$ and $(1, ab^{-1})$, you can identify $a$ from the intercept and then solve for $b$ using the second point. The process is identical — only the exponent sign changes.

6. Comparing Growth and Decay: Side by Side

Placing a growth curve and a decay curve on the same axes makes the contrast clear. Both share the same $y$-intercept when $a$ is the same, and both approach $y = 0$ as an asymptote — but in opposite directions along the $x$-axis.

7. Practice and Exam Technique

You have now covered all the key ideas. Before looking at common mistakes, try these practice problems to consolidate the method. Work through each one using the sketch steps from earlier in the lesson.

Examiners testing exponential graphs tend to focus on three areas: the $y$-intercept, the asymptote, and the correct direction of the curve. Getting all three right earns full marks on a sketch question. Here are the mistakes that cost students marks most often.