GCSE · Maths · AQA · Spec 8300 · Foundation

Recognise, sketch, interpret linear and quadratic graphs

Here's a secret most people miss: you can know exactly what a graph looks like before you plot a single point. The equation gives it away.

Take a walk along y = x² − 4x − 5

x: 2.5. y: -8.75

Drag the point along the curve (or use the arrow keys) and hunt for four things. 1) Where does it cross the y-axis? Go to x = 0. 2) Where does it cross the x-axis? Find the places where y = 0 — there are two. 3) Where is the lowest point? Nudge left and right until y stops falling and starts rising. 4) Now stand at x = 0, then at x = 4. What do you notice about y?

Watch out: The x-axis is the line y = 0, and the y-axis is the line x = 0. On this grid they are the lines through the 0 labels — not the edges of the frame.

Watch it done: sketching a parabola

Problem

Sketch the graph of y = x² − 2x − 3 for x from −2 to 4.

Name the shape before you draw it

Pick an equation, then put it in the family you think it belongs to. Look at the highest power of x — or whether x is on the bottom of a fraction.

Still to sort

Linear: a straight line (0)

x appears only as x (power 1).

Where the line is: A minus sign or a number in front of x never bends the line. y = 5 − 2x is still straight — it just slopes downwards.

Quadratic: a parabola (U or ∩) (0)

The highest power of x is x².

Where the line is: A minus sign in front of x² turns the U upside down into a ∩. It is still a parabola.

Cubic: an S-bend (0)

The highest power of x is x³.

Where the line is: Extra lower-power terms, such as + x, don't change the family. The x³ is in charge.

Reciprocal: two separate branches (0)

x is on the bottom of a fraction.

Where the line is: Only x on the BOTTOM counts. y = x/2 has x on top, so it is just a straight line.

9 of 9 still to sort.

Here's the trick: ignore the numbers in front and the plus-or-minus bits, and look only at what happens to x. That one detail decides the whole shape.

The strange one: y = 1/x

Sam has worked out these points for y = 1/x: (−2, −0.5), (−1, −1), (−0.5, −2), (0.5, 2), (1, 1) and (2, 0.5). The points are plotted. Now Sam has to join them up.

Which is closest to what you think the finished graph looks like?

It goes through (0, 0), like lots of graphs do.It's one curve: you join (−0.5, −2) to (0.5, 2) straight across the middle.Join each point to the next with a ruler, so it looks neat.Two separate smooth curves, one top-right and one bottom-left, and neither touches an axis.

How sure are you?

I'm sureI think soI'm guessing

It goes through (0, 0), like lots of graphs do.

This is a common misconception

Try putting x = 0 into y = 1/x. You get 1 ÷ 0, and that has no answer at all — you can't share 1 into 0 equal groups. So there is no point on the graph where x = 0: it can't go through the origin, and it can't touch the y-axis anywhere.

It's one curve: you join (−0.5, −2) to (0.5, 2) straight across the middle.

This is a common misconception

Look at what happens as x creeps towards 0. At x = 0.1, y = 10. At x = 0.01, y = 100. The graph shoots up the side of the y-axis instead of crossing it. From the negative side it shoots down: x = −0.1 gives y = −10. There's no bridge across x = 0, so the two halves never join.

Join each point to the next with a ruler, so it looks neat.

This is a common misconception

The graph isn't just your six dots — it's every point that fits the equation. Check halfway between (1, 1) and (2, 0.5). A ruler line gives y = 0.75 at x = 1.5, but 1 ÷ 1.5 = 0.666…, so the real graph is lower. Straight pieces put you in the wrong place, so draw smooth curves.

Two separate smooth curves, one top-right and one bottom-left, and neither touches an axis.

This matches the evidence

Spot on. x = 0 is banned (1 ÷ 0 has no answer), so there's a gap at the y-axis. And 1 divided by any number is never 0, so the graph never reaches the x-axis either. Positive x gives positive y (top-right), negative x gives negative y (bottom-left), and each branch hugs the axes more and more closely without ever touching them.

Two smooth branches, top-right and bottom-left, never touching either axis — because x can't be 0 and y can never be 0.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Know a graph's shape from its equation, sketch a parabola properly, and read what its features tell you.

What you need to know

The equation tells you the shape: x (power 1) gives a straight line, x² gives a parabola, x³ gives a cubic, and y = 1/x gives two separate branches.
A quadratic graph is a smooth U, or a ∩ when there's a minus in front of the x². It has exactly one turning point and a vertical line of symmetry through it.
The y-intercept is where x = 0. The x-intercepts (roots) are where y = 0 — a parabola can have two, one or none.
To sketch a quadratic: make a table of values, plot the points, join them with one smooth curve, then label the key features.
y = 1/x has no value at x = 0 and is never 0, so its graph never touches either axis.

The big picture

Every graph is simply all the points that fit its equation. Look at what happens to x and you know the family: x on its own gives a straight line, x² gives a parabola, x³ gives an S-bend cubic, and x on the bottom of a fraction (y = 1/x) gives two separate branches. A quadratic is a smooth, symmetrical U (or ∩) with one turning point, and its features — where it crosses the axes and where it turns — are things you can read straight off the graph.

Key points

1Highest power of x: 1 → straight line, 2 → parabola, 3 → cubic. x on the bottom of a fraction (y = 1/x) → reciprocal curve.

2A minus in front of x² flips the parabola upside down, so its turning point becomes a highest point instead of a lowest one.

3Read a graph's features as facts about the equation: crossing the y-axis means x = 0; crossing the x-axis means y = 0.

4A parabola is symmetrical: points at the same height sit the same distance either side of the turning point.

5Join plotted points with a smooth curve, never with ruler segments — the graph includes every point in between.

6Square negative numbers inside brackets: (−3)² = 9, not −9.

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Worked example

Problem

Without drawing it accurately, describe the graph of y = 8 + 2x − x²: its shape, where it crosses the axes, and where it turns.

⚠ Watch out

Squaring a negative x wrongly. (−3)² = (−3) × (−3) = +9, but writing −3² means −(3²) = −9. Get this wrong in a table of values and the left-hand side of your parabola dives downwards instead of rising — so always bracket the negative before you square it.

🧠

Memory hook

Look at x's biggest power. 1 → a line. 2 → a U-turn. 3 → an S-bend. x on the bottom → split in two.

✓

Check yourself

Without plotting anything: which way up is the parabola y = x² − 9, and where does it cross the y-axis? (Answer: the right way up, a U, crossing the y-axis at (0, −9).)

Flashcards

(10)

How can you tell from its equation that a graph will be a parabola?

The highest power of x is x² (a quadratic). Its graph is a smooth U, or a ∩ if the x² term is negative.

What does a minus sign in front of x² do to a parabola?

It flips it upside down: a ∩ instead of a U, so the turning point is a highest point (a maximum).

What is the turning point of a parabola?

The single point where the curve changes direction: the lowest point of a U, or the highest point of a ∩.

Where is the line of symmetry of a parabola?

The vertical line through its turning point. The curve is a mirror image either side of it.

How do you find where any graph crosses the y-axis?

Put x = 0 into the equation. The y you get is the y-intercept.

What do the points where a graph crosses the x-axis have in common?

Their y-coordinate is 0. For a quadratic they are called the roots, and there can be two, one or none.

Why does the graph of y = 1/x never touch the y-axis?

The y-axis is where x = 0, and 1 ÷ 0 has no value — so there is no point there.

Why does the graph of y = 1/x never touch the x-axis?

The x-axis is where y = 0, and 1 divided by any number is never 0.

What does the graph of y = x³ look like?

An S-bend: it climbs from the bottom-left, flattens out through (0, 0), then climbs to the top-right.

How should you join plotted points when sketching a curve?

With one smooth freehand curve through the points — never straight ruler segments, and a rounded turning point rather than a sharp V.

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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Take a walk along y = x² − 4x − 5

Name the shape before you draw it

The strange one: y = 1/x

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