2.6PureStretch

Modelling with Quadratics

Quadratics model anything with a single turning point — the path of a thrown ball, the area of a field, profit against price. The maths is the same as before; the new skill is translating a real situation into a quadratic and interpreting the answer.

30 min Video by Zeeshan Zamurred Quadratics

Edexcel AS level Maths: 2.6 Modelling with QuadraticsWatch the full walkthrough before the notes below.

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What you'll be able to do

Set up a quadratic model from a worded problem
Use the turning point to find a maximum or minimum
Interpret roots and intercepts in context
Decide which solutions are physically sensible

From words to a quadratic

Define your variables clearly, then write the relationship as a quadratic. Projectile-motion problems are the classic example, where height depends on time.

h = - 5 t^{2} + u t + h_{0}

A typical height model:

u

is the launch speed and

h_{0}

the start height.

Maximum and minimum values

Because a quadratic has exactly one turning point, completing the square (or the line of symmetry) finds the greatest or least value of the model — the maximum height, minimum cost, and so on.

1Complete the square:

- 5 (t^{2} - 4 t) = - 5 [(t - 2)^{2} - 4] = - 5 (t - 2)^{2} + 20

2The maximum of

h

is the constant

20

, reached when

t = 2

Answermaximum height

20

m at

t = 2

Tip — A maximum/minimum question is really a “find the vertex” question in disguise.

Interpreting the roots

In context, the roots and intercepts have real meanings. For a height-time model, a root is when the object is at height zero — usually when it lands.

1Set

h = 0

- 5 t (t - 4) = 0

2So

t = 0

(launch) or

t = 4

Answerit lands at

t = 4

Choosing sensible answers

A quadratic gives two solutions, but the context often rules one out. Negative times, negative lengths or values beyond the model’s range should be discarded with a brief reason.

Tip — Always sanity-check: a time of −2 s or a width of −3 m is not physically possible, so reject it.

Formula recap

h = - 5 t^{2} + u t + h_{0}

Standard projectile-height model.

max/min at the vertex (- p, q)

From completed-square form.

h = 0 \Rightarrow object at ground level

Roots give contextual events.

Common mistakes to avoid

Keeping both algebraic solutions even when one is impossible.

Reject solutions that make no physical sense (e.g. negative time) and say why.

Confusing the time of maximum height with the maximum height itself.

The vertex gives both: the t-coordinate is when, the y-coordinate is how high.

Key takeaways

Define variables, then translate the situation into a quadratic.
The vertex gives the maximum or minimum value of the model.
Roots and intercepts carry real-world meaning (e.g. landing time).
Discard solutions that are not physically sensible.

Test yourself

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