KS3 · Maths

Equations and their graphs (linear and non-linear)

Read an equation like y = 3x + 2 and know its graph before plotting a single point. Straight or curved? How steep? Where does it cross? Here's how.

Lift the dot. The line pivots.

Gradient m 2

Gradient m: 2. change in y for +1 in x: 2.0

This is y = mx + c with c = 1. The dot sits one step right of the y-axis, at x = 1, so the number beside it is the change in y for that one step: exactly m. Lift the dot above the height of (0, 1) and the line rises; the higher you lift, the steeper the climb. Drop it below and the line falls. However you move it, the line never lets go of (0, 1), where it crosses the y-axis. That point belongs to c, not m.

Watch out: A negative gradient isn't a mistake on the graph. It means that every +1 in x takes y down. At m = −2, each step right drops the line by 2.

Your turn: plot a graph from its equation

Build the table for y = 2x − 5

Plot the graph of y = 2x − 5 for 0 ≤ x ≤ 5.

The range 0 ≤ x ≤ 5 tells you which x values to use: x = 0, 1, 2, 3, 4 and 5.Without a range, the straight line y = 2x − 5 would carry on forever in both directions. The range says which piece of it to draw.
x = 0: y = 2 × 0 − 5 = −5, so the first point is (0, −5).
missing step

Which line is step 3?

x = 1: y = 2 × 1 − 5 = −3, so the point is (1, −3).x = 1: y = 2 × 1 − 5 = 3, so the point is (1, 3).x = 1: y = 21 − 5 = 16, so the point is (1, 16).

Predict, then check

Work out a few y values in your head first: try x = −3, −1, 0, 1 and 3.

You make a table of values for y = x² from x = −3 to x = 3 and plot the points. What will the graph look like?

Spot the shape from the equation

Line, parabola, or neither?

Which shape will each equation's graph make?

Still to sort

Straight line (0)

x appears only to the power 1 (plain x).

Where the line is: The order doesn't matter: 9 − 2x is still x to the power 1, so it's still a line.

Parabola (0)

The highest power of x is 2. That's a quadratic.

Where the line is: A minus sign in front of x² flips the bowl upside down, but it's still a parabola.

Neither (0)

The highest power of x is more than 2.

Where the line is: x³ is a higher power than 2, so it isn't a quadratic and its graph isn't a parabola.

8 of 8 still to sort.

Find the highest power of x in each equation. That one number decides the shape.

Which do you believe?

Where does the 3 put the line?

Here's a line written a slightly awkward way round: y = 3 − ⅓x.

Which is closest to what you think right now?

The 3 means the line crosses the x-axis at (3, 0).The gradient is 3, because the first number in the equation is the gradient.The gradient is ⅓, and the line crosses the y-axis at (0, 3).It crosses the y-axis at (0, 3), and the gradient is −⅓.

How sure are you?

I'm sureI think soI'm guessing

The 3 means the line crosses the x-axis at (3, 0).

This is a common misconception

Test it. Put x = 3 into the equation: y = 3 − ⅓ × 3 = 3 − 1 = 2. So the point is (3, 2), not on the x-axis at all. The constant c is where the line crosses the y-axis. Put x = 0 in and you get y = 3, so it crosses at (0, 3). (It does meet the x-axis eventually, at (9, 0): 3 − ⅓ × 9 = 0.)

The gradient is 3, because the first number in the equation is the gradient.

This is a common misconception

The gradient isn't decided by position. It's the number multiplying x, wherever that term is written. Here the x term is −⅓x, so the gradient is −⅓. The 3 on its own, with no x attached, is c.

The gradient is ⅓, and the line crosses the y-axis at (0, 3).

This is a common misconception

You've got the crossing point right, but the minus sign belongs to the x term. The gradient is −⅓, so the line falls: every +1 in x takes y down by a third. Leave the sign behind and you'd draw the line sloping the wrong way.

It crosses the y-axis at (0, 3), and the gradient is −⅓.

This matches the evidence

Exactly. c = 3 gives the y-intercept (0, 3), and the number multiplying x, −⅓, is the gradient. It's the same line as y = −⅓x + 3, just written the other way round.

In y = mx + c, c is where the line crosses the y-axis, at (0, c). The gradient is the number in front of x, minus sign included, wherever it is written.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Read an equation and know its graph before you plot a single point.

What you need to know

To plot a graph, make a table of values: substitute each x value to find its y, then plot the coordinates and join them.
If the highest power of x is 1, the graph is a straight line. If the highest power is 2, it's a quadratic and the graph is a parabola.
In y = mx + c, m is the gradient: the change in y for every +1 in x. Positive m rises, negative m falls.
In y = mx + c, c gives the y-intercept: the line crosses the y-axis at (0, c).
Gradient = change in y ÷ change in x, moving in the positive x direction. Lines with the same gradient are parallel.

The big picture

Different types of equation make different shapes of graph. To plot one, make a table of values: substitute each x, find its y, plot the points and join them. When the highest power of x is 1, y changes by the same amount for every +1 in x: a straight line. When the highest power is 2, the equation is a quadratic and its graph is a symmetrical curve called a parabola. For a line y = mx + c, m is the gradient (the change in y for every +1 in x) and c gives the y-intercept, (0, c). Lines with the same gradient are parallel.

Key points

1A straight line has a constant difference: y changes by the same amount for every +1 in x. For y = 2x − 5 that change is +2; for y = 4 − x it is −1.

2The gradient shows up in three places: as the constant difference in the table, as the steepness of the graph, and as the number multiplying x in the equation.

3The y-intercept is where x = 0. So y = 3x + 2 crosses the y-axis at (0, 2).

4y = x² is symmetrical about the y-axis because a negative number squared is positive. Join its points with a smooth curve, never with straight lines.

5A parabola that opens downwards, like y = 3 − x², is still a parabola. y = 2x³ makes a cubic curve, which is not a parabola.

6A sketch has no scale and isn't exact, but it shows the sign of the gradient, which line is steeper, and the y-intercept, labelled.

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Worked example

Problem

A straight line goes through the points (0, 1) and (4, 4). Find its gradient and its equation.

⚠ Watch out

Dividing the wrong way round. For the points (1, 2) and (3, 8), y changes by 6 and x changes by 2, so the gradient is 6 ÷ 2 = 3, not 2 ÷ 6 = 1/3. Change in y always goes on top.

🧠

Memory hook

Power 1? A line. Power 2? A bowl. And in y = mx + c, m is how much it climbs or drops per step right, while c is where it crosses: up the y-axis, never along the x.

✓

Check yourself

Before plotting y = 4x − 5: line or curve, rising or falling, and where does it cross the y-axis? (A straight line rising steeply, gradient +4, crossing the y-axis at (0, −5).)

Flashcards

(13)

How do you make a table of values for an equation like y = 3x + 1?

Choose some x values, substitute each one into the equation to find its y, and write down each pair as a coordinate (x, y) ready to plot.

How can a table of values tell you a graph will be a straight line?

y changes by the same amount for every +1 in x. That constant difference means the points lie in a straight line.

Which power of x gives a straight-line graph?

An x exponent of one (plain x), as in y = 3x − 2. These are linear equations.

What is a quadratic equation, and what shape is its graph?

An equation where the highest power of x is 2, such as y = x² + x. Its graph is a parabola.

Why is the graph of y = x² symmetrical about the y-axis?

A negative number squared is positive, so x and −x give the same y. For example, (−3)² = 9 and 3² = 9.

How should you join the plotted points of y = x²?

With a smooth curve. The points aren't in a straight line, so straight-line joins give the wrong shape.

What does the gradient of a line measure?

How steep the line is: the change in y for every change of +1 in x.

In y = mx + c, what do m and c tell you?

m is the gradient. c gives the y-intercept, the point (0, c) where the line crosses the y-axis.

Is the gradient always the first number in a line's equation?

No. The gradient is the number multiplying x, sign included, wherever it is written: y = 5 − 3x has gradient −3.

How do you find a gradient from two points on a line?

Change in y ÷ change in x, moving in the positive x direction. The answer can be a fraction.

What can you say about two lines with the same gradient?

They are parallel. For example, y = 3x + 2 and y = 3x − 2.

What must a sketch of a straight line show?

The right direction of slope (sign of the gradient), steeper lines looking steeper, and the y-intercept labelled. It has no scale, so it is only approximate.

Why do car headlights use parabola-shaped reflectors?

With the bulb at the parabola's focus, the light reflects off in a parallel, focused beam.

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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Equations and their graphs (linear and non-linear)

Lift the dot. The line pivots.

Build the table for y = 2x − 5

Line, parabola, or neither?

Where does the 3 put the line?

Read the features off each equation

More KS3 Maths topics