GCSE · Maths · AQA · Spec 8300 · Higher

Turning points by completing the square (Higher)

Where does a quadratic graph turn round? You could plot dozens of points to find out — or rewrite the algebra so that it simply tells you.

Slide p — and watch the valley run the other way

-4-2024-11357xy
p, the number in (x + p)² 0

p, the number in (x + p)²: 0. x-coordinate of the turning point: 0.0

Line of symmetry. The red line is x = −p. It passes through the turning point, and the curve is a mirror image on either side of it — so it moves whenever p does.

This is y = (x + p)² + 1. Push p up to 2 and the bottom of the curve moves LEFT, to x = −2. Pull p down to −2 and it moves right, to x = 2. The height of the bottom never changes: it stays at y = 1.

Why the turning point is at (−p, q)

?

Reason it through

Why does y = a(x + p)² + q turn at x = −p, at height q — and why is the curve symmetrical?

Link 1 of 4

First link · your turn

Can (x + p)² ever be negative?

2
Locked — reveal the link above first
3
Locked — reveal the link above first
4
Locked — reveal the link above first

Getting into completed square form

Where does the extra square come from?

Quadratics rarely arrive in completed square form, so we have to build it. Take x² − 6x + 5. Halve the −6 to get −3 and try (x − 3)². Fill in the grid to see what (x − 3)² really gives you.

Grid: each cell is its row times its column
x−3
xx²
−3

Type x² as x^2 if you cannot type ². Spaces do not matter.

When a isn't 1

Problem

Write 2x² + 12x + 7 in the form a(x + p)² + q, and find the turning point of y = 2x² + 12x + 7.

Your turn

Fill the gaps: a negative a

Write −3x² + 12x − 5 in the form a(x + p)² + q, and find the turning point of y = −3x² + 12x − 5.

  1. −3x² + 12x − 5
  2. missing step
Which line is step 2?

Reading the graph without drawing it

Twice, once or never?

How many times does each graph meet the x-axis?

Still to sort

Meets it twice (0)

From the turning point, the curve heads towards the axis and crosses it on both sides.

Where the line is: A minimum below the axis, or a maximum above it.

Touches it once (0)

The turning point sits exactly on the x-axis: a repeated root.

Where the line is: q = 0, whether it is a minimum or a maximum.

Never meets it (0)

From the turning point, the curve heads away from the axis on both sides.

Where the line is: A minimum above the axis, or a maximum below it.

6 of 6 still to sort.

For each graph, find the turning point, decide whether it is a minimum or a maximum, then picture which way the curve goes from there.

Watch out: A turning point below the x-axis does not always mean two crossings. A maximum below the axis never reaches it.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

y = a(x + p)² + q

The turning point, the line of symmetry, valley or hill — all sitting right there in the numbers.

What you need to know

  • How the numbers in y = a(x + p)² + q give you the turning point, its line of symmetry and whether it is a minimum or maximum.
  • Why the turning point is where the bracket equals zero.
  • How to complete the square for x² + bx + c, including when the number in the bracket is negative.
  • How to complete the square when the x² term has a coefficient other than 1.
  • How to tell from the turning point whether a graph meets the x-axis twice, once or not at all.

The big picture

Completing the square rewrites a quadratic as a(x + p)² + q. In that form the turning point is (−p, q), the line of symmetry is x = −p, and the sign of a tells you whether the turning point is a minimum or a maximum. From the turning point you can also tell how many times the graph meets the x-axis.

Key points

1y = a(x + p)² + q has its turning point at (−p, q) and its line of symmetry at x = −p.
2a > 0 gives a minimum (a valley); a < 0 gives a maximum (a hill).
3The turning point is at x = −p because a square is never negative, so (x + p)² is at its smallest, zero, when x = −p.
4x² + bx + c = (x + b/2)² − (b/2)² + c. The (b/2)² is always subtracted, even when b/2 is negative.
5For ax² + bx + c, factor a out of the x² and x terms first — don't divide by it. In general, ax² + bx + c = a(x + b/(2a))² + c − b²/(4a).
6A minimum below the x-axis, or a maximum above it, meets the axis twice; a turning point on the axis touches it once (a repeated root); otherwise the graph never meets it.

Worked example

Problem

Write x² + 5x + 1 in the form (x + p)² + q. Hence state the turning point of y = x² + 5x + 1 and its line of symmetry.

⚠ Watch out

Giving the turning point of y = (x + 3)² − 7 as (3, −7). The x-coordinate is the value that makes the bracket zero, which is x = −3, so the turning point is (−3, −7). The y-coordinate keeps its sign: it is q exactly as written.

🧠

Memory hook

The bracket wants to be zero — so x does the opposite of what the bracket says. q is the height. And a is the mood: positive smiles (a valley), negative frowns (a hill).

✓

Check yourself

In your head: where does y = 4(x − 7)² − 3 turn? Minimum or maximum? Line of symmetry? (Answer: (7, −3); a minimum; x = 7.)

Flashcards

(13)
What is the completed square (turning point) form of a quadratic graph?
y = a(x + p)² + q
Where is the turning point of y = a(x + p)² + q?
At (−p, q): x is the value that makes the bracket zero, and q is the height.
How can you tell whether the turning point of y = a(x + p)² + q is a minimum or a maximum?
From the sign of a: a > 0 gives a minimum, a < 0 gives a maximum.
What is the line of symmetry of y = a(x + p)² + q?
The vertical line x = −p, through the turning point.
Why is a(x + p)² at its least (a > 0) or greatest (a < 0) when x = −p?
A square is never negative, and (x + p)² is zero only when x = −p.
Why is the graph of y = a(x + p)² + q symmetrical?
At x = −p + k and x = −p − k the bracket is k and −k, and k² = (−k)², so both points have the same y.
Complete the square: x² + bx + c = ?
(x + b/2)² − (b/2)² + c
Why do you subtract (b/2)² when completing the square?
Expanding (x + b/2)² gives x² + bx + (b/2)², which is (b/2)² too much.
If the number in the bracket is negative, do you add or subtract its square?
Subtract it, always. Squaring a negative number gives a positive one.
What is the first step in completing the square for ax² + bx + c when a ≠ 1?
Factor a out of the x² and x terms, e.g. 4x² + 8x + 1 = 4(x² + 2x) + 1.
Why don't you divide an expression by a when completing the square?
Dividing changes the expression. You can divide both sides of an equation, because they stay equal, but not an expression on its own.
ax² + bx + c in completed square form is…?
a(x + b/(2a))² + c − b²/(4a)
In what three ways can a quadratic graph meet the x-axis?
Twice; touching it once (a repeated root, at the turning point); or not at all (no real roots).

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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