⚙️ Function Kingdom · Functions

Exponential Functions

Model repeated growth and decay with $y = a * b x$ , and contrast constant differences with constant ratios.

In short

Exponential functions multiply by a constant factor each step: $y = a * b x$ , with a the value at $x = 0$ .
Constant differences in a table mean linear; constant ratios mean exponential.
A p% increase multiplies by $1 + p/100$ each period; a p% decrease multiplies by 1 - $p/100$ .
Doubling and half-life problems are solved by counting steps first, then converting steps into time.

Multiplying, not adding

A linear function adds the same amount at each step. An exponential function multiplies by the same amount at each step.

$y = a * b x$

Here a is the value at $x = 0$ (the starting amount) and b is the constant multiplier, called the base or growth factor.

$y = 3 * 2 x x 0 1 2 3 4 y 3 6 12 24 48$

Each output is double the one before. Compare with the linear $y = 3 + 2x$ , which gives $3, 5, 7, 9, 11$ . At first the linear function is ahead; very soon the exponential one leaves it far behind. Exponential growth always overtakes linear growth in the end.

Remember that the exponent applies only to b. In $3 * 2 4$ the 3 is a multiplier, so the value is $3 x 16 = 48$ , not $64$ .

Reading a table

Given a table, two tests settle what kind of relationship it is.

Subtract neighbouring outputs. If the differences are constant, it is linear.
Divide neighbouring outputs. If the ratios are constant, it is exponential.

$x 0 1 2 3 y 5 15 45 135$

Differences: $10, 30, 90$ — not constant. Ratios: $3, 3, 3$ — constant. So the relationship is exponential with $b = 3$ , and since $y = 5$ when $x = 0$ , the model is $y = 5 * 3 x$ .

Do both tests before deciding. Checking only one of them is how a table gets misclassified.

Growth and decay by percentage

Percentage change is exponential, because the percentage is taken of the current amount, not the original one.

Growing by 8% multiplies by 1.08 each period (you keep the original 100% and add 8%).
Shrinking by 8% multiplies by 0.92 each period (100% minus 8%).

So 500 gold growing at 8% per year is

$y = 500 * 1.08 x$

After 3 years that is $500 x 1.259712 = 629.86$ gold. Adding 8% of the original 500 three times would give only 620 — the difference is the interest earned on the interest.

When $b > 1$ the model grows; when $0 < b < 1$ it decays. The percentage itself is never the multiplier: 8% gives 1.08 or 0.92, never 0.08.

Doubling and halving times

Many exponential processes are described by how long one doubling (or halving) takes.

"A slime colony doubles every 3 hours, starting at 40 slimes. When does it reach 640?"

Count doublings first, not hours: $640 / 40 = 16$ , and $16 = 2 4$ , so four doublings are needed. Each takes 3 hours, so the answer is 12 hours.

Half-life works identically. A potion that halves every 5 days and drops from 96 units to 12 has gone through three halvings $(96 -> 48 -> 24 -> 12)$ , so 15 days have passed.

The habit to build is: convert the ratio of amounts into a number of steps, then convert steps into time. Reporting the number of doublings as if it were the time is a very common slip.

Worked examples

Example 1

f(x) $= 4 * 3 x$ . Find f(3).

The exponent belongs only to the 3, so evaluate $33$ first.
$33 = 3 x 3 x 3 = 27$ .
Now multiply by the starting amount: $4 x 27$ .
$f(3) = 108$ .

Example 2

A village of 2000 people grows by 10% per year. What is the population after 2 years?

Growing by 10% means multiplying by $1 + 10/100 = 1.1$ each year.
After 2 years the model gives $2000 x 1.1 2$ .
$1.1 2 = 1.21$ .
$2000 x 1.21 = 2420$ people. (Adding 200 twice would wrongly give 2400.)

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

f(x) $= 4 * 2 x$ Find f(3).

Answer: 32

$f(3) = 4 x 2 3$
$23 = 8$
$f(3) = 4 x 8 = 32$

Problem 2

Difficulty 3 of 5

A magical lantern in Hugo's pack starts with 1,000 units of light and loses 50% of its light every hour. How much is left after 4 hours? Round to 2 decimal places.

Answer: 62.5 units

Multiplier per hour: 1 - $50/100 = 0.5$
Amount $= 1,000 x 0.5 4$
$0.5 4 = 0.0625$
Amount $= 62.5$

Problem 3

Difficulty 4 of 5

A potion loses half its strength every 8 days. It starts at 32 units. After how many days is it down to 1 unit?

Answer: 40 days

$32 -> 16 -> 8 -> 4 -> 2 -> 1$
That is 5 halvings.
Time $= 5 x 8 = 40$ days

Common mistakes

Applying the exponent to the whole product, reading $3 * 2 4$ as $64$ .
Treating percentage change as linear by taking the percentage of the original amount every time.
Using the percentage itself (0.08) as the multiplier instead of 1.08 or 0.92.
Giving the number of doublings as the answer when the question asked for the elapsed time.

What you should be able to do

Evaluate an exponential function at a given input.
Identify growth and decay from the base.
Write an exponential model from a starting amount and a rate.
Compare linear and exponential growth over time.

Where this fits in the curriculum

Common Core

HSF-LE.A.1.A
High school — Prove that linear functions grow by equal differences and exponential functions by equal factors over equal intervals.
HSF-LE.A.1.C
High school — Recognise situations in which a quantity grows or decays by a constant percent rate per unit interval.
HSF-LE.A.2
High school — Construct an exponential function given a graph, a description of a relationship, or a table of values.
8.F.A.3
Grade 8 — Interpret y = mx + b as a linear function and give examples of functions that are not linear.

Ontario

MTH1W.C3.1
Grade 9 de-streamed — Compare the shapes of the graphs of linear and non-linear relations to describe their rates of change.
This is as far as Ontario goes before Grade 11: MTH1W contrasts linear with non-linear growth but never names an exponential function — that is MCR3U and MHF4U.

SAT

Passport to Advanced Math
Exponential expressions, functions and their graphs.
Problem Solving and Data Analysis
Growth and decay models expressed as a constant percent rate.
The SAT splits exponential work: the symbolic side sits in Passport to Advanced Math, the "population grows 4% a year" side in Problem Solving and Data Analysis.