GCSE · Maths · AQA · Spec 8300 · Foundation

Roots, intercepts and turning points of quadratics — graphically

Every quadratic graph has a mirror line down the middle, and that mirror tells you exactly where the curve turns.

Slide along y = x² − 4x − 5 and let the coordinates do the talking

x: 2.5. y: -8.75

Drag the point along the curve. Hunt for three things: where y reads 0 (the roots), where x reads 0 (the y-intercept), and the lowest point, where y stops falling and starts rising (the turning point). Then try a mirror test: compare x = 0 with x = 4, and x = −2 with x = 6.

Watch out: The y-axis is the line x = 0, the grid line labelled 0 at the bottom. It isn't the left-hand edge of this picture. Slide until x reads 0 to find the y-intercept.

Use the mirror

Find the turning point's x without drawing anything

Each letter is a different quadratic curve. Using only the facts given, place the x-coordinate of its turning point on the line. Then check.

A — Curve A has roots at x = −6 and x = 2B — Curve B has roots at x = 1 and x = 6C — Curve C passes through (−5, 4) and (3, 4)

Maths · Algebra

Now find those roots with algebra

Same curve as the graph above. Step through and watch the x-axis crossings fall out of the working.

GoalFind the roots of y = x² − 4x − 5 without looking at the graph

x² − 4x − 5 = 0

y = 0

Roots are where the curve meets the x-axis, and every point on the x-axis has y = 0. So set the expression equal to 0.

Step 1 of 4

Roots are where the curve meets the x-axis, and every point on the x-axis has y = 0. So set the expression equal to 0.

Watch out: The number in the bracket has the opposite sign to the root: (x + 1) = 0 gives x = −1, not x = 1.

Higher

Higher: completing the square

Problem

Write x² − 4x − 5 in the form (x − a)² + b, and use it to find the turning point of y = x² − 4x − 5.

What do you really think?

Four students, one curve

Four students are describing the graph of y = x² − 6x + 8.

Which student is right?

“It has a root at 8. That's the number on its own, so it's where the curve crosses the axis.”“Its turning point is (0, 8). The curve turns where it meets the y-axis.”“(Higher) It can be written as (x − 3)² − 1, so its turning point is (−3, −1).”“Its roots are 2 and 4, so its turning point has x-coordinate 3.”

How sure are you?

I'm sureI think soI'm guessing

“It has a root at 8. That's the number on its own, so it's where the curve crosses the axis.”

This is a common misconception

8 is where the curve crosses the y-axis: put x = 0 and you get y = 8, so the y-intercept is (0, 8). Roots are on the x-axis, where y = 0. Solve x² − 6x + 8 = 0: (x − 2)(x − 4) = 0, so the roots are x = 2 and x = 4.

“Its turning point is (0, 8). The curve turns where it meets the y-axis.”

This is a common misconception

A curve only turns on the y-axis if the y-axis happens to be its line of symmetry. This curve's roots are 2 and 4, so its mirror line is x = 3. At x = 0 the curve is still going down, and it keeps going down until x = 3.

“(Higher) It can be written as (x − 3)² − 1, so its turning point is (−3, −1).”

This is a common misconception

The form is right, but the x-coordinate isn't. (x − 3)² equals 0 when x = +3, not −3. So the turning point is (3, −1). The sign inside the bracket flips; the number outside doesn't.

“Its roots are 2 and 4, so its turning point has x-coordinate 3.”

This matches the evidence

Yes. Factorising x² − 6x + 8 = 0 gives (x − 2)(x − 4) = 0, so the roots are 2 and 4. Halfway between them is 3. Substitute x = 3 to get y = 9 − 18 + 8 = −1, so the turning point is (3, −1).

Each feature has its own condition. Roots: y = 0. y-intercept: x = 0. Turning point: on the line of symmetry, halfway between the roots.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One symmetrical curve, three key points. Find two of them and you can work out the third.

What you need to know

Roots are where the curve meets the x-axis, so y = 0 there. Usually the curve crosses the axis, but it can just touch it.
The y-intercept is where the curve crosses the y-axis, so x = 0 there. For y = ax² + bx + c it is (0, c).
The turning point is where the curve changes direction: its lowest point, or its highest point if the curve is upside down.
A quadratic graph is symmetrical. The turning point lies on the line of symmetry, halfway between the roots.
Higher: completing the square to get y = (x − a)² + b shows the turning point is (a, b).

The big picture

A quadratic graph has three key features. The roots are where it meets the x-axis (y = 0): usually it crosses, but it can just touch the axis and turn back. The y-intercept is where it crosses the y-axis (x = 0). The turning point is where it changes direction. Because the curve is symmetrical, the turning point lies on the line of symmetry, halfway between the roots. You can find roots with algebra by setting y = 0 and factorising. At Higher, completing the square, y = (x − a)² + b, gives the turning point (a, b) directly.

Key points

1To find roots algebraically, set y = 0, factorise, then set each bracket equal to 0.

2Turning point x-coordinate = (first root + second root) ÷ 2. Substitute that x to get the y-coordinate.

3Any two points at the same height are mirror images. The line of symmetry is halfway between their x-values.

4Higher: in (x − a)² + b the bracket is 0 when x = a, so the sign inside the bracket flips but b does not.

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Worked example

Problem

For the curve y = x² + 2x − 8, find the y-intercept, the roots and the turning point.

⚠ Watch out

Mixing up the roots and the y-intercept. The constant at the end of y = x² + bx + c is where the curve crosses the y-axis. It is not a root. Roots are on the x-axis, where y = 0, and you find them by solving the equation.

🧠

Memory hook

Fold the U in half. The two roots land on top of each other, and the crease runs straight through the turning point.

✓

Check yourself

For y = x² − 8x + 12, find the roots, the y-intercept and the turning point. (Answer: roots x = 2 and x = 6; y-intercept (0, 12); turning point (4, −4).)

Flashcards

(8)

What is a root of a quadratic graph?

An x-value where the curve crosses (or touches) the x-axis, so y = 0 there.

How do you find the y-intercept of y = ax² + bx + c?

Put x = 0. Everything with an x vanishes, leaving y = c, so the intercept is (0, c).

What is the turning point of a quadratic graph?

The point where the curve changes direction: the minimum of a U-shaped curve, or the maximum of an upside-down one.

A quadratic has roots p and q. Where is its line of symmetry?

x = (p + q) ÷ 2, halfway between the roots. The turning point lies on this line.

Two points on a quadratic curve have the same y-value. What does that tell you?

They are mirror images of each other, so the line of symmetry is halfway between their x-values.

How do you find the roots of a quadratic algebraically?

Set y = 0, factorise, then set each bracket equal to 0 and solve.

Higher: what is the turning point of y = (x − a)² + b?

(a, b). The x-coordinate has the opposite sign to the number inside the bracket.

Higher: why is the lowest value of y = (x − a)² + b equal to b?

A square can't be negative, so (x − a)² is at least 0. It equals 0 when x = a, leaving y = b.

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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Slide along y = x² − 4x − 5 and let the coordinates do the talking

Find the turning point's x without drawing anything

Now find those roots with algebra

Four students, one curve

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