Exponential Functions

Draw and read graphs of y = a to the power x: tables of values, growth and decay, the asymptote, solving from the graph, and real growth problems.

In the exam for Cambridge O Level Maths 4024, Cambridge IGCSE Maths 0580 · 16 real past-paper questions below

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What an Exponential Graph Is

In $y = 2^{x}$ , x is the power. Each time x goes up by one, y is multiplied by 2.

x	0	1	2	3
y	1	2	4	8

Rule

Every graph $y = a^{x}$ passes through $(0, 1)$ , because any number to the power 0 is 1.

Key idea

The curve gets closer and closer to the x-axis on the left but never touches it. That line is called an asymptote.

Key idea

Puzzle: paper is about $0.1$ mm thick, and each fold doubles it. If you could fold a sheet 42 times, how thick would it be? Answer: $0.1 \times 2^{42}$ mm is about 440 000 km. That is farther than the Moon, which is about 384 400 km away!

Completing the Table

The most common question: fill in a table for a curve like $y = 3 \times 2^{x}$ . For each x, work out the power first, then multiply by 3.

Worked example

1
$x = - 2 : y = 3 \times 2^{- 2}$ — Put x = −2 into the equation. The power is negative.
2
$2^{- 2} = \frac{1}{2 ^{2}} = \frac{1}{4}$ — A negative power means one over: flip it to 1 over $2^{2}$ .
3
$y = 3 \times \frac{1}{4} = 0.75$ — Now multiply by 3. A quarter of 3 is 0.75.
4
$x = 0 : 2^{0} = 1$ — Any number to the power 0 is 1.
5
$y = 3 \times 1 = 3$ — Multiply by 3. This is where the curve crosses the y-axis.
6
$x = 0.5 : 2^{0.5} = 1.414 \dots$ — A decimal power: use your calculator's power key. Keep all the digits for now.
7
$y = 3 \times 1.414 \dots = 4.24$ — Multiply, then round only at the end. Tables usually want 2 decimal places.
8
$x = 3 : 2^{3} = 8$ — Power first: 2 × 2 × 2 = 8.
9
$y = 3 \times 8 = 24$ — Then multiply by 3.

Common mistake

Writing $2^{- 2} = - 4$ . A negative power means one over: $2^{- 2} = \frac{1}{4}$ . The y values are never negative.

Key idea

Drawing the graph next is worth 3 or 4 marks, and almost all of them are for plotting the points accurately. Plot every pair from your table, then join them with one smooth curve, not straight lines with a ruler.

Growth or Decay

The number being raised to the power tells you the shape. Bigger than 1: the curve rises. Between 0 and 1: it falls.

Worked example

1
$y = (\frac{1}{2})^{x}$ — The base is one half. It is between 0 and 1, so this is decay: the curve falls.
2
$x = 0 : y = (\frac{1}{2})^{0} = 1$ — Any number to the power 0 is 1, so the curve still passes through (0, 1).
3
$x = 1 : y = \frac{1}{2} = 0.5$ — One step right means multiplying by one half, so y halves.
4
$x = 2 : y = \frac{1}{4} = 0.25$ — Another step right: y halves again.
5
$x = - 1 : y = (\frac{1}{2})^{- 1} = 2$ — A negative power flips the fraction: one over a half is 2. Going left, y doubles.
6
$x = - 2 : y = (\frac{1}{2})^{- 2} = 2^{2} = 4$ — Flip the half to 2, then square it: 4.

Common mistake

Thinking a decay curve reaches 0. It halves forever and only gets close to the x-axis.

Sketching the Curve

Some questions just say “sketch”. You get blank axes and no table. The marks are for the right shape and the right crossing point.

Rule

A sketch of $y = a^{x}$ needs three things: the curve on both sides of the y-axis, the point $(0, 1)$ labelled, and the curve staying above the x-axis, getting close on one side but never touching it.

Worked example

1
$y = 2^{- x}$ — The curve to sketch. The minus in the power decides the shape.
2
$2^{- x} = \frac{1}{2 ^{x}} = (\frac{1}{2})^{x}$ — A negative power means one over, so this is the same curve as $y = (\frac{1}{2})^{x}$ .
3
$base \frac{1}{2} < 1$ — The base is between 0 and 1, so the curve falls from left to right: decay.
4
$x = 0 : y = 2^{0} = 1$ — Label $(0, 1)$ on the y-axis. Every curve $y = a^{x}$ crosses there.
5
$x large : y close to 0$ — On the right the curve gets close to the x-axis but never reaches it, because $2^{- x}$ is never 0 or negative.

Common mistake

Letting the curve cross the x-axis, or drawing it on one side of the y-axis only. Each of these loses a mark.

Shifted Up: Adding a Number

Adding a number moves the whole curve up. The asymptote and the y-intercept move with it.

Worked example

1
$y = 2^{x} + 3$ — The curve. The + 3 is added after the power, so every point moves up 3.
2
$x = 0 : y = 2^{0} + 3$ — To find the y-intercept, put x = 0.
3
$y = 1 + 3 = 4$ — $2^{0} = 1$ , then add 3. The curve crosses the y-axis at (0, 4).
4
$x = - 10 : 2^{- 10} \approx 0.001$ — Try a very negative x. The power part is almost 0.
5
$y \approx 0.001 + 3$ — So y is just above 3, and gets even closer to 3 the further left you go.
6
$asymptote: y = 3$ — The curve never reaches 3, so the line y = 3 is the asymptote.

Rule

For $y = a \times b^{x} + c$ : the y-intercept is $a + c$ and the asymptote is $y = c$ .

Common mistake

Saying the curve crosses the y-axis at 3. Put x = 0 in: $2^{0} + 3 = 4$ .

Finding the Unknown Numbers

Some questions give points on the curve and ask for the missing numbers. Start with x = 0: the power becomes 1, so one unknown disappears.

1. $y = a^{x} + b$ passes through $(0, 3)$ and $(2, 18)$ . Find a and b.

Worked example

1
$3 = a^{0} + b$ — Put in the point (0, 3): x = 0 and y = 3.
2
$3 = 1 + b$ — $a^{0} = 1$ , because any number to the power 0 is 1. Now a has gone.
3
$b = 2$ — Take 1 from both sides.
4
$18 = a^{2} + 2$ — Now put in (2, 18) and the b you just found.
5
$a^{2} = 16$ — Take 2 from both sides.
6
$a = 4$ — Square root. −4 also squares to 16, but reject it: the base of an exponential must be positive.
7
$asymptote: y = 2$ — The number added on the end, b, gives the asymptote.

2. $y = c^{x}$ passes through $(3, 125)$ . Find y when x = −1.

Worked example

1
$125 = c^{3}$ — Put in the point (3, 125).
2
$c = 3125 = 5$ — Cube root both sides. Check: 5 × 5 × 5 = 125.
3
$x = - 1 : y = 5^{- 1}$ — Now use $y = 5^{x}$ with the new x.
4
$y = \frac{1}{5} = 0.2$ — A negative power means one over.

Key idea

Write down each equation after you substitute, like $18 = a^{2} + 2$ . That line earns a mark for working even if a later number goes wrong.

Solving From the Graph

To solve $3 \times 2^{x} = 10$ with the graph, draw the line $y = 10$ and read down from the crossing.

Worked example

1
$3 \times 2^{x} = 10$ — The equation. The left side is exactly the curve on your grid, $y = 3 \times 2^{x}$ .
2
$y = 10$ — So draw the flat line y = 10 across the grid.
3
$crossing at (1.7, 10)$ — Find where the line meets the curve.
4
$x \approx 1.7$ — Read straight down to the x-axis. The answer is this x value.
5
$3 \times 2^{1.7} \approx 9.7$ — Check on a calculator: close to 10, so the reading is sensible.

Common mistake

Giving 10 as the answer. The answer is the x value of the crossing point.

Make It Match the Graph

Often the equation is not exactly the curve you drew. Change it, doing the same to both sides, until one side is the curve. The other side is the line to draw.

Worked example

1
$curve drawn: y = 2^{x} - 4$ — The curve on your grid.
2
$2^{x} = 7$ — The equation to solve. It has $2^{x}$ but not the − 4.
3
$2^{x} - 4 = 7 - 4$ — Take 4 from both sides, so the left side matches the curve.
4
$2^{x} - 4 = 3$ — 7 − 4 = 3.
5
$y = 3$ — Draw this flat line on the grid.
6
$x \approx 2.8$ — Read down from where the line crosses the curve.

Key idea

If the power is x + 2, split it first: $2^{x + 2} = 2^{x} \times 2^{2} = 4 \times 2^{x}$ . Then multiply or divide both sides until you reach the curve's equation.

Common mistake

Drawing y = 7. That line solves $2^{x} - 4 = 7$ , a different equation. Change both sides first.

Growth and Decay in Real Life

Word questions use a formula like $N = 200 \times 2^{t}$ . The number in front is the start amount, when t is 0.

Worked example

1
$N = 200 \times 2^{t}$ — 200 bacteria at the start, doubling every hour. t is the number of hours.
2
$t = 3 : N = 200 \times 2^{3}$ — After 3 hours, so put t = 3.
3
$2^{3} = 8$ — Power first: 2 × 2 × 2 = 8.
4
$N = 200 \times 8 = 1600$ — Then multiply by the start amount: 1600 bacteria.
5
$M = 80 \times 0. 5^{t}$ — 80 mg of medicine, halving every hour. A base below 1 means decay.
6
$t = 3 : 0. 5^{3} = 0.125$ — Power first: 0.5 × 0.5 × 0.5 = 0.125.
7
$M = 80 \times 0.125 = 10$ — 10 mg is left after 3 hours.

Common mistake

Working out $200 \times 2$ first and then cubing. Always do the power before multiplying.

Which Method? Read the Question

Look at where x is. If x is the power, it is this lesson.

You see	You do
Complete the table for $y = 3 \times 2^{x}$	power first, then multiply
Growth or decay?	base above 1 rises, below 1 falls
State the asymptote	the number added on the end
Use the graph to solve $3 \times 2^{x} = 10$	draw the flat line, read x
Sketch $y = 2^{- x}$	falling curve through (0, 1), above the x-axis
Find a and b from points on the curve	put x = 0 first: the power gives 1
Solve $2^{x} = 7$ on the graph of $y = 2^{x} - 4$	change it to match the curve, then a flat line
Solve $2^{x} = x + 3$ with a line	a sloped line: Solving Equations by Graphical Methods
Estimate the gradient at a point	draw a tangent: Gradient of a Curve at a Point
Solve exactly, like 2 to the power x is 32	Exponential Equations
Next term: multiply each time	Exponential Sequences
Money growing by a percentage	Compound Interest

Key idea

Check a table: each y value should be the one before multiplied by the same number.

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