← LearnYear 9 Maths · QuadraticsFree placement check
Year 9AC9M9A05

Modelling with Linear and Quadratic Functions

Use mathematical modelling to solve applied problems involving change including financial contexts; formulate problems, choosing to use either linear or quadratic functions; interpret solutions in terms of the situation; evaluate the model and report methods and findings. Experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns.

Choosing the right model is the real skill. A constant rate of change is linear. A changing rate is quadratic.

Builds onYr 9 · Solving Quadratic Equations

Method

Constant change means linear
Equal steps up for equal steps across.
A changing rate means quadratic
Area, projectiles and anything squared.
Test the second differences
Constant first differences mean linear. Constant second differences mean quadratic.
Changing $a$ stretches, changing $c$ shifts
One parameter at a time. Note what moves and what does not.
Always check the model against reality
A negative length or time means the model does not apply there.

Worked example

e.g. A ball’s height is h=20t−5t2. Find when it lands and its greatest height.\text{A ball's height is } h = 20t - 5t^2. \text{ Find when it lands and its greatest height.}
  1. Landing means the height is zero.20t−5t2=020t - 5t^2 = 0
  2. Factorise.5t(4−t)=05t(4 - t) = 0
  3. Solve.t=0  or  t=4t = 0 \ \text{ or } \ t = 4
  4. Reject the one that makes no sense. t=0t = 0 is the throw.it lands at t=4 s\text{it lands at } t = 4 \text{ s}
  5. The greatest height sits midway between the roots.t=2t = 2
  6. Substitute back.h=20(2)−5(2)2=20 mh = 20(2) - 5(2)^2 = 20 \text{ m}

Practice

Decide the model first, then solve, then check it makes sense.

1
A table has first differences 3,3,3. Linear or quadratic?\text{A table has first differences } 3,3,3. \text{ Linear or quadratic?}
AnswerLinear.\text{Linear.}
2
First differences 2,4,6. Second differences?\text{First differences } 2,4,6. \text{ Second differences?}
Answer2,2 — quadratic2,2 \text{ — quadratic}
3
In y=ax2, what does increasing a do?\text{In } y = ax^2, \text{ what does increasing } a \text{ do?}
AnswerMakes the parabola narrower.\text{Makes the parabola narrower.}
4
In y=x2+c, what does c do?\text{In } y = x^2 + c, \text{ what does } c \text{ do?}
AnswerShifts it up or down.\text{Shifts it up or down.}
5
h=30t−5t2. When does it land?h = 30t - 5t^2. \text{ When does it land?}
Answert=6 st = 6 \text{ s}
6
Your model gives a negative time. What now?\text{Your model gives a negative time. What now?}
AnswerReject it. The model does not apply there.\text{Reject it. The model does not apply there.}
Next step
Practise Modelling with Linear and Quadratic Functions with instant marking
A free 10-minute placement check finds which Year 9 topics to work on first, then Summit builds a weekly plan around them.
See where my child is →