Building quadratic models from information | Year 11

01 / LEARN

The idea, step by step

Choose the form first

Known roots suggest factored form; a known turning point suggests vertex form; three general points can determine expanded form. Substitute the remaining condition to find the coefficient.

Define variables and units before calculating. A mathematical root may represent a negative time or an impossible length, so only accept it after checking the physical situation.

Interpret and check

A model only describes its stated domain and assumptions. A quadratic arch between supports need not describe a structure beyond those supports. A simple projectile model ignores effects such as air resistance.

Check every given point or condition. A vertex gives the greatest or least value only if its x-coordinate lies in the allowed domain. Explain both vertex coordinates in context.

A common trap

Reporting both quadratic roots as physical answers.

Instead: Apply restrictions from the defined time, length or other variable.

02 / WORKED EXAMPLES

See the thinking

Example 1

A symmetric arch

An arch is at ground level at x=0 m and x=8 m. At x=4 m its height is 6 m. Find its rule.

Use $h = a x (x - 8)$ .
Substitute (4,6): $6 = - 16 a$ , so a=-3/8.
Restrict the domain to the span.

h = - \frac{3}{8} x (x - 8)

0 \leq x \leq 8

Example 2

Greatest rectangular area

A rectangle has perimeter 28 cm. Find its maximum area.

For width x, the length is 14-x.
Area is $x (14 - x) = - {(x - 7)}^{2} + 49$ .
The vertex x=7 lies within 0<x<14.

49 square centimetres, at dimensions 7 cm by 7 cm.

Check your understanding

A parabola has vertex (2,5) and passes through (0,1). Find its rule.

Show answer

$y = - {(x - 2)}^{2} + 5$

In vertex form, 1=4a+5, hence a=-1.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Roots are -2,4 and the graph passes through (0,-16). Find its rule.
3 marks
Vertex is (3,-2) and the graph passes through (1,6). Find its rule.
3 marks
A rectangle has perimeter 40 cm. Find maximum area and dimensions.
3 marks
A ball has height $h = - 5 t^{2} + 20 t + 1$ m. Find maximum height/time.
3 marks
A rectangle’s length is 3 cm more than its width; area is 40 square centimetres. Find dimensions.
3 marks
An arch has supports x=-3 m, x=3 m and maximum height 9 m. Give its rule/domain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Roots are -2,4 and the graph passes through (0,-16). Find its rule.
3 marks
Use factored form.
At x=0, -16=-8a.
$y = 2 (x + 2) (x - 4)$
Vertex is (3,-2) and the graph passes through (1,6). Find its rule.
3 marks
Use vertex form.
Substitute (1,6): 6=4a-2.
$y = 2 {(x - 3)}^{2} - 2$
A rectangle has perimeter 40 cm. Find maximum area and dimensions.
3 marks
Area is $x (20 - x) = - {(x - 10)}^{2} + 100$ .
The vertex is within 0<x<20.
100 square centimetres; 10 cm by 10 cm.
A ball has height $h = - 5 t^{2} + 20 t + 1$ m. Find maximum height/time.
3 marks
Complete the square: $h = - 5 {(t - 2)}^{2} + 21$ .
Interpret the vertex.
21 m at t=2 s.
A rectangle’s length is 3 cm more than its width; area is 40 square centimetres. Find dimensions.
3 marks
Use $x (x + 3) = 40$ .
Factor $(x + 8) (x - 5) = 0$ and reject -8.
5 cm by 8 cm.
An arch has supports x=-3 m, x=3 m and maximum height 9 m. Give its rule/domain.
3 marks
Use $h = a (x + 3) (x - 3)$ .
At x=0, 9=-9a.
$h = 9 - x^{2}$ , $- 3 \leq x \leq 3$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

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