Modelling with Functions & Transformations
Learn modelling with functions and transformations for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-02). Real data rarely fits a basic graph exactly, so you pick a suitable family of functions — linear, quadratic, exponential or trigonometric — and adjust it until it matches.
You will learn to apply translations, dilations and reflections using the form \(y = af(b(x + c)) + d\), interpret what each constant means in context, and use the model to answer practical questions — the everyday work of anyone who turns data into predictions.
Theory
Modelling with functions uses a transformed standard function to describe a real situation. This Year 12 Mathematics Advanced topic (MAV-12-02) uses quadratic models in vertex form \(h=a(t-p)^2+q\) for a maximum or minimum, and exponential models \(P=Ab^{\,t}\) for growth and decay, without calculus.
A quadratic model in vertex form \(h=a(t-p)^2+q\) has its turning point at \((p,q)\): a maximum if \(a<0\), a minimum if \(a>0\), with value \(q\) at \(t=p\).
An exponential model \(P=Ab^{\,t}\) starts at \(A\) when \(t=0\); it grows if \(b>1\) and decays if \(0
Method
- Choose the model: quadratic for a turning point, exponential for a constant percentage change.
- Read or substitute: vertex \((p,q)\); value at a time by substituting \(t\).
- Solve for a time by setting the model equal to the target value.
| \(\text{vertex}\) | \(=\) | \((2,20)\) |
| \(\text{max}\) | \(=\) | \(20\text{ m at }t=2\text{ s}\) |
| \(5(t-2)^2\) | \(=\) | \(20\) |
| \((t-2)^2\) | \(=\) | \(4\) |
| \(t\) | \(=\) | \(4\ (t\ge0)\) |
| \(P\) | \(=\) | \(200(1.1)^3\) |
| \(=\) | \(200\times1.331\approx266\) |
| \(P(0)\) | \(=\) | \(50\) |
| \(P(3)\) | \(=\) | \(50\times8=400\) |
| \(2^{\,t}\) | \(=\) | \(16\Rightarrow t=4\) |
Common pitfalls
Frequently asked questions
How do you find the maximum of a quadratic model?
Write it in vertex form h = a(t - p) squared + q. The turning point is at (p, q), so if a is negative the maximum value is q, reached at t = p.
What does the vertex of a parabola tell you in a model?
The vertex gives the best or worst value of the model, such as the greatest height of a projectile or the lowest cost, and the time at which it occurs.
How do you find the starting value of an exponential model?
Substitute t = 0. In P = A times b to the t, the starting value is A because b to the power 0 is 1.
How do you tell if an exponential model grows or decays?
It grows when the base b is greater than 1 and decays when b is between 0 and 1.