7. Modelling with Functions | Mathematics Advanced

Starting point: Years 9 and 10

Direct variation

$y$ is directly proportional to $x$ when $y = k x$ . The graph is a straight line through the origin. $k$ is the constant of variation.

Variation can also be with a power: the stopping distance of a car varies with the square of its speed, $d = k v^{2}$ . Double the speed and the distance becomes four times as long.

Inverse variation

$y$ is inversely proportional to $x$ when $y = \frac{k}{x}$ . Double one, the other halves.

Reading a model

A model has parameters (the fixed numbers) and variables. In $C = 0.5 n + 20$ for Chloe's cost, 20 is the fixed cost and 0.5 is the cost per cup.

Modelling

Variation

Direct: y = kx
Power: y = kx²
Inverse: y = k/x

Pick a shape

Constant rate: linear
Max or min: quadratic
Grows by a percent: exponential
Repeats: trigonometric

Graphs solve equations

Intersection points

Check

Sensible domain
Compare with data

Year 12Choosing and fitting a model

Look at the pattern in the data to pick the type of function:

Pattern	Model
Same amount added each step	linear $y = m x + c$
Rises then falls, or a best value	quadratic
Same percentage change each step	exponential $y = A e^{k t}$
Repeats in cycles	trigonometric $y = a sin (b (t - c)) + d$

Example. Chloe finds that at price $p$ dollars she sells $n = 100 - 20 p$ cups. Profit is

$P = (p - 0.5) (100 - 20 p) - 20$

This is a quadratic that opens downward. The product $(p - 0.5) (100 - 20 p)$ is zero at $p = 0.5$ and $p = 5$ , and taking away 20 does not move the vertex sideways. So the best price is halfway: $p = 2.75$ dollars. Calculus gives the same result by setting $P^{'} (p) = 0$ .

Using transformations to fit. Daily temperature can be modelled by a sine curve: the centre line is the average, the amplitude is half the difference between high and low, and the period is 24 hours. Each parameter has a real meaning.

Quick summary

Direct variation: $y = k x$ . Inverse variation: $y = \frac{k}{x}$ . Find $k$ from one known pair.
A model has parameters with real meanings, such as a fixed cost and a rate.
Pick the function type from the pattern: constant rate, maximum, percentage growth or cycles.
Solutions of $f (x) = g (x)$ are where the two graphs meet.
Always check that predictions make sense and state a sensible domain.

Quiz

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