Modelling with Functions

In short: Modelling with functions means choosing a function that matches a real situation, like a straight line for a constant rate or $y = \frac{k}{x}$ for inverse variation. Find the constants from known values, then check that the model's predictions make sense.

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.

Inverse variation can use a power too. Light intensity varies inversely with the square of the distance: $I = \frac{k}{d ^{2}}$ . If $I = 36$ at 2 m, then $k = 36 \times 4 = 144$ , and at 6 m, $I = \frac{144}{36} = 4$ . Tripling the distance makes the light 9 times weaker.

The same method works for any power, such as $W = k r^{3}$ : find $k$ from the known pair, then use the formula. Put every measurement in the same units first, for example $5000 cm^{2} = 0.5 m^{2}$ (because $1 m^{2} = 10000 cm^{2}$ ).

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.

Fitting a quadratic through three points. Put each point into $y = a x^{2} + b x + c$ to get three equations. Through $(0, 1)$ , $(1, 4)$ and $(2, 5)$ : $c = 1$ , $a + b + 1 = 4$ and $4 a + 2 b + 1 = 5$ . So $a + b = 3$ and $2 a + b = 2$ , giving $a = - 1$ , $b = 4$ , and $y = - x^{2} + 4 x + 1$ . Its greatest value is at $x = 2$ .

Piecewise models. Some prices change rule part way: a hire costs 30 dollars for up to 3 hours, then 10 dollars for each extra hour. Write a rule for each part ( $C = 30$ for $0 < t \leq 3$ , and $C = 30 + 10 (t - 3)$ for $t > 3$ ), then compare with other plans part by part.

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

13 questions with instant marking. It is free, and your scores are saved to your account so you can carry on from any device.

Turn on JavaScript to sign in and take the quiz.

Try 3 questions now

$y$ varies directly with $x$ , and $y = 12$ when $x = 3$ . Find $y$ when $x = 5$ .
1. A 15
2. B 20
3. C 36
4. D 7.2
Show answer
B: 20
$k = 4$ , so $y = 4 \times 5 = 20$ .
The graph of a direct variation $y = k x$ is:
1. A a straight line through the origin
2. B a parabola
3. C a hyperbola
4. D a horizontal line
Show answer
A: a straight line through the origin
$y = k x$ has gradient $k$ and intercept 0.
$d = k v^{2}$ . If $v$ doubles, $d$ becomes:
1. A double
2. B half
3. C four times as big
4. D the same
Show answer
C: four times as big
$(2 v)^{2} = 4 v^{2}$ .