GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Roots, intercepts and turning points of quadratics

Find the point where a quadratic turns and you already know how many times it crosses the x-axis, before any algebra. Grab the lowest point below and lift it.

Lift the lowest point. Watch the roots.

Vertical shift -1

Vertical shift: -1. turning point’s y: -1.00

The red line is the x-axis, y = 0. The roots are the x values where the curve meets it: two crossings, one touch, or none at all.

This is y = (x − 3)² + q, where q is how far the curve has been shifted up or down. While the turning point sits below the x-axis the curve crosses twice. At q = 0 the two crossings meet and the curve just touches at x = 3. Any higher and it never reaches the axis. At q = −1, 0 and 1 the curve is y = x² − 6x + 8, y = x² − 6x + 9 and y = x² − 6x + 10.

Watch out: A quadratic does not always have two roots. Lift this curve 1 unit at a time and it goes from two roots, to one repeated root, to none.

Why the turning point is (−p, q)

Reason it through

Why is the lowest point of y = (x − 3)² − 1 exactly at (3, −1)?

Link 1 of 4

First link · your turn

What is the smallest value a squared number can ever be?

Locked — reveal the link above first

Maths · Algebra

Complete the square, then read everything off

Step through it. Each line tells you what was done and why.

GoalFind the turning point and the roots of y = x² − 14x + 40

y = x² − 14x + 40

start

The number to watch is the x coefficient, −14.

Step 1 of 8

The number to watch is the x coefficient, −14.

Your turn

When x² comes with a minus sign

Find the turning point of y = 6x − x². Is it a maximum or a minimum?

y = 6x − x²The x² term is negative, so the graph is an upside-down U (∩). It will have a highest point.
missing step

Which line is step 2?

y = −(x² + 6x)y = −(x² − 6x)y = x² − 6x

Spot the slip

The factorising route

Find the y-intercept, the roots and the turning point of y = x² − 2x − 15.

A student’s answer — which line goes wrong?

y-intercept: put x = 0, so y = 0 − 0 − 15. It crosses at (0, −15).
Roots: set y = 0, so x² − 2x − 15 = 0.
Factorise: (x + 3)(x − 5) = 0
So x = 3 or x = −5
Turning point is halfway between the roots: x = (3 + (−5)) ÷ 2 = −1
y = (−1)² − 2(−1) − 15 = −12, so the turning point is (−1, −12)

When it won’t factorise

No tidy brackets. So what about the roots?

y = 2x² − 9x + 8 won’t factorise: no pair of whole-number brackets multiplies out to it. Drawn accurately, its graph is a U shape with its lowest point below the x-axis.

What can you say about the roots of y = 2x² − 9x + 8?

It won’t factorise, so it has no roots.Its roots are where it meets the y-axis, at (0, 8).Its roots are where it crosses the x-axis, about 1.2 and 3.3.

How sure are you?

I'm sureI think soI'm guessing

It won’t factorise, so it has no roots.

This is a common misconception

Factorising only works when the roots are tidy numbers. The roots are where the graph crosses the x-axis, and this graph does cross it, twice. The numbers just aren’t whole.

Its roots are where it meets the y-axis, at (0, 8).

This is a common misconception

That’s the y-intercept, where x = 0. Roots are where y = 0, on the x-axis. A root is also an x value, not a point: you’d write x ≈ 1.2, not (0, 8).

Its roots are where it crosses the x-axis, about 1.2 and 3.3.

This matches the evidence

Yes. Look where the accurate graph crosses the x-axis and read off the x values: x ≈ 1.2 and x ≈ 3.3. They are estimates, so the accuracy of your drawing matters.

Won’t factorise doesn’t mean no roots. If the curve crosses the x-axis, you can estimate the roots from an accurate graph, as x values.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

y = (x + p)² + q

Three features to find on one curve: where it meets the y-axis, where it crosses the x-axis, and where it turns.

What you need to know

The y-intercept is where x = 0.
The roots are the x values where y = 0, where the graph crosses or touches the x-axis.
y = (x + p)² + q has its turning point at (−p, q).
A quadratic can have two roots, one repeated root or no roots.

The big picture

A quadratic graph is a smooth U or ∩ shape. It meets the y-axis where x = 0, and its roots are the x values where it crosses or touches the x-axis, where y = 0. The turning point is its lowest or highest point: writing y = (x + p)² + q puts it at (−p, q). Where that turning point sits decides whether there are two roots, one repeated root or none.

Key points

1A quadratic graph is a smooth U shape when the x² term is positive and a ∩ shape when it is negative. Join plotted points with a smooth curve, not straight lines.

2y-intercept: put x = 0. It is the constant term, so y = x² + 3 meets the y-axis at (0, 3).

3Roots: set y = 0 and solve. If it factorises, set each bracket equal to 0. If it doesn’t, estimate the x values where an accurate graph crosses the x-axis.

4Turning point: complete the square to get y = (x + p)² + q, and the turning point is (−p, q). If you know both roots, its x-coordinate is halfway between them.

5If the x² coefficient isn’t 1, take it out first: 6x − x² = 9 − (x − 3)², a maximum at (3, 9).

6For a U shape: minimum below the x-axis gives two roots, on the axis gives one repeated root, above the axis gives none. For a ∩ shape it flips: a maximum below the axis, like the one on y = −2x² + x − 8, means no roots.

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Worked example

Problem

For y = x² + 4x − 5, find the y-intercept, the turning point and the roots.

⚠ Watch out

Getting the sign of the turning point wrong. In y = (x − 3)² − 1 the bracket is 0 when x = +3, so the turning point is (3, −1), not (−3, −1). The same trap catches roots: x + 3 = 0 gives x = −3.

🧠

Memory hook

Low point under the line? Two roots. Touching the line? One. Floating above it? None.

✓

Check yourself

Without expanding anything, can you say how many roots y = (x + 1)² + 4 has, and explain why using its turning point?

Flashcards

(15)

What is the y-intercept of a quadratic graph, and how do you find it?

Where the graph meets the y-axis. Put x = 0: for y = x² + bx + c it is (0, c).

What is a root of a quadratic?

An x value where the graph crosses or touches the x-axis, so y = 0. Write it as x = …, not as a point on the y-axis.

How do you find the roots when the quadratic factorises?

Set y = 0, factorise, then set each bracket equal to 0. (x + 3)(x − 5) = 0 gives x = −3 or x = 5.

How do you find the roots when it won’t factorise?

Draw an accurate graph and estimate the x values where it crosses the x-axis.

Where is the turning point of y = (x + p)² + q?

At (−p, q). The square is 0 when x = −p, which leaves y = q.

Why can y = (x + p)² + q never go below q?

Because a square is never negative, so (x + p)² adds 0 or more on to q.

How do you complete the square on x² + bx + c?

Write (x + b/2)², then subtract (b/2)², the extra number that bracket brings in, and keep the c: x² + bx + c = (x + b/2)² − (b/2)² + c.

You know both roots. Where is the turning point’s x-coordinate?

Exactly halfway between them. Roots x = −3 and x = 5 give x = 1.

What shape is the graph when the x² term is negative?

An upside-down U (∩), with a maximum turning point instead of a minimum.

First step to complete the square when the x² coefficient isn’t 1?

Take it out as a factor: 6x − x² = −(x² − 6x). Then complete the square inside the bracket.

How does a U-shaped curve’s turning point tell you the number of roots?

Minimum below the x-axis: two roots. On the axis: one repeated root. Above the axis: none.

When does a ∩-shaped quadratic have no roots?

When its maximum turning point is below the x-axis, so even its highest point never reaches the axis.

What is a repeated root?

The two roots are the same x value because the turning point sits on the x-axis. y = (x − 3)² touches the axis only at x = 3.

Square-rooting to find roots: what must you remember?

Both signs. (x − 7)² = 9 gives x − 7 = 3 or x − 7 = −3.

Why join the plotted points of a quadratic with a smooth curve?

Points in between don’t lie on straight lines. On y = x², x = ½ gives y = ¼, below the ruler line from (0, 0) to (1, 1).

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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Lift the lowest point. Watch the roots.

Complete the square, then read everything off

When x² comes with a minus sign

The factorising route

No tidy brackets. So what about the roots?

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