The Parabola: y = a(x + p)² + q (Grade 11)

The turning-point form of the parabola, what a, p and q each do to the graph, finding intercepts and the axis of symmetry, and writing down domain and range.

1Turning-point form

y = a(x + p)² + q

Turning point: (−p ; q)
Axis of symmetry: x = −p

⚠️

The turning point is (−p ; q), not (p ; q). In y = 2(x + 2)² − 8 we have p = 2, so the turning point is at x = −2.

2What each parameter does

a — a > 0 opens upwards (minimum); a < 0 opens downwards (maximum). A bigger |a| makes it narrower.
p — shifts left or right. Positive p shifts left.
q — shifts up or down, and is the y-value of the turning point.

y = 2(x + 2)² − 8, showing the turning point and both x-intercepts.

3Finding the intercepts

Worked ExampleSketch y = 2(x + 2)² − 8

1
a = 2 > 0, so it opens upwards.
Shape first.
2
Turning point: (−2 ; −8).
(−p ; q).
3
y-intercept: let x = 0 → y = 2(2)² − 8 = 0.
So it passes through the origin.
4
x-intercepts: let y = 0 → 2(x + 2)² = 8 → (x + 2)² = 4.
5
x + 2 = ±2, so x = 0 or x = −4.
Do not forget the ± — it gives the second intercept.

4Domain and range

Domain: x ∈ ℝ (always)
Range: y ≥ q if a > 0 · y ≤ q if a < 0

For y = 2(x + 2)² − 8 the range is y ≥ −8, because the graph opens upwards from its lowest point.

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Range questions are about the turning point and nothing else. Ask: does it open up or down, and what is q?

5Converting from the standard form

Worked ExampleWrite y = x² − 6x + 5 in turning-point form

1
Halve the coefficient of x: −6 ÷ 2 = −3, then square it: 9.
Completing the square.
2
y = (x² − 6x + 9) − 9 + 5.
Add 9 and subtract 9 — the value is unchanged.
3
y = (x − 3)² − 4.
4
Turning point (3 ; −4).
Here p = −3, so −p = 3.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Write down the turning point of y = 2(x + 2)² − 8.
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(−2 ; −8)
2
Determine the x-intercepts of y = 2(x + 2)² − 8.
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(x+2)² = 4 → x + 2 = ±2 → x = 0 or x = −4
3
Determine the y-intercept of y = 2(x + 2)² − 8.
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x = 0 → y = 8 − 8 = 0
4
Write down the equation of the axis of symmetry of y = 2(x + 2)² − 8.
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x = −2
5
Write down the range of y = 2(x + 2)² − 8.
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y ≥ −8
6
Write down the domain of y = 2(x + 2)² − 8.
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x ∈ ℝ
7
Write y = x² − 6x + 5 in the form y = a(x + p)² + q.
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y = (x − 3)² − 4
8
For what values of x is y = 2(x + 2)² − 8 increasing?
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To the right of the turning point: x > −2