KS3 · Maths

Gradient and y-intercept of a linear graph

Which is steeper: a line with gradient 2, or a line with gradient −5? Most say 2, because −5 is the smaller number. Drag the dot above and test it.

Tilt the line. Read the step.

Gradient m 1

Gradient m: 1. change in y per step of 1: 1.0

Step of one. The red stretch runs from x = 1 to x = 2, exactly one step to the right. How far the line climbs or drops across it is the gradient. If it does neither, the gradient is 0.

Lift the dot to change the gradient m. Every line here is y = mx + 2, so each one pivots about the same point, (0, 2). The red stretch always covers exactly one step to the right.

Maths · Reading a graph

Where does the working go wrong?

A line passes through the points (1, 3) and (2, 5). On this grid, each square across is worth 0.5 on the x axis and each square up is worth 1 on the y axis. A student works out the gradient. Which line of their working is the first to go wrong?

A student's working — which line goes wrong?

The gradient is how much y changes for a step of one to the right.
I start at the point (1, 3), which sits exactly on a grid corner.
I draw an arrow one square to the right, because that is my step of one.
Then I go up until I meet the line again. That is 1 square up, so y changes by 1.
One step right, one up: the gradient is 1.

Maths · Gradient without a graph

Your turn to supply the steps

Find the gradient of the line through (−7, 3) and (3, −2), without drawing it.

Rough sketch first: (−7, 3) is on the left and higher up, (3, −2) is on the right and lower down.
missing step

Which line is step 2?

So the line falls as it goes right, and the gradient will be negative.So the line rises as it goes right, and the gradient will be positive.So the gradient will be negative, because one of the coordinates is negative.

Maths · The y intercept

Where does y = x + 2 cross the y axis?

The line y = x + 2 is drawn on a pair of axes. It crosses the y axis at one point.

Which idea is closest to what you think right now?

The y axis is where x is zero. Put x = 0 in: y = 0 + 2 = 2, so it crosses at (0, 2).It crosses where y is zero: x + 2 = 0, so x = −2.It is just the number 2.

How sure are you?

I'm sureI think soI'm guessing

The y axis is where x is zero. Put x = 0 in: y = 0 + 2 = 2, so it crosses at (0, 2).

This matches the evidence

Yes. Every point on the y axis has x = 0, so the y intercept is wherever the line has x = 0. It is a point, so you write it as a coordinate: (0, 2). The same move works on other equations. For x + y = 10, x = 0 gives (0, 10). For 2x − y = −3, x = 0 gives 0 − y = −3, so y = 3 and the point is (0, 3). Tidy those into y = −x + 10 and y = 2x + 3 and the number on the end is the y value of that point.

It crosses where y is zero: x + 2 = 0, so x = −2.

This is a common misconception

Putting y = 0 finds where the line meets the x axis, which is the point (−2, 0). That is a different place. The y axis is where x is zero, so x is the one to set to 0.

It is just the number 2.

This is a common misconception

Right place, wrong form. An intercept is a point where the line meets an axis, so it needs both coordinates: (0, 2). The 2 is only its y value.

For the y intercept, put x = 0 in and write a coordinate like (0, 2). A vertical line has no y intercept.

Predict, then check

The table skips x = 0. Work out how y changes for each step of 1 first.

A straight line has these values: x = 8, 9, 10, 11 and y = 2, 5, 8, 11. Where does it cross the y axis?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

How steep a straight line is, which way it leans, and where it crosses the y axis.

What you need to know

Gradient means one thing: how much y changes when x goes up by exactly 1.
A line that rises to the right has a positive gradient. A line that falls to the right has a negative gradient.
Steepness goes by the size of the gradient once you ignore the sign, so −5 is steeper than 2.
Parallel lines have exactly the same gradient. Being equally steep is not enough.
The y intercept is the coordinate where the line crosses the y axis, where x = 0, written like (0, 2).

The big picture

The gradient of a straight line is how much y changes when x goes up by exactly one. Its sign tells you whether the line rises or falls, and its size (ignoring the sign) tells you how steep it is. The y intercept is the point where the line crosses the y axis, where x = 0. You can find both from graphs, tables, equations and pairs of points.

Key points

1Gradient = the change in y for a step of one in the positive x direction.

2Positive gradient: y increases as you move right. Negative gradient: y decreases.

3Larger absolute value means steeper. Gradients 1 and −1 are equally steep.

4Between two points: smaller x first, find the change in x and the change in y, then divide to a change in x of 1. Keep exact fractions.

5The y intercept is found by putting x = 0 in, or by counting back to x = 0, and is written as a coordinate. A vertical line has none.

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Worked example

Problem

Find the gradient and the y intercept of the line through (4, 10) and (7, 16).

⚠ Watch out

Counting grid squares as if each one were a step of one. Read the scales on the axes and move a real distance of 1 along the x axis.

🧠

Memory hook

Gradient: go ONE across, then see how far y climbs or drops. y intercept: ZERO is the x that matters.

✓

Check yourself

Without drawing anything: a line has gradient −4 and another has gradient 3. Which line is steeper, which one falls, and what would you need to know about them to say they are parallel?

Flashcards

(15)

What is the gradient of a straight line?

The amount y changes when x goes up by exactly one, a step of one to the right.

How can you tell the sign of a gradient from a line?

If y increases as the line goes right, the gradient is positive. If y decreases as the line goes right, it is negative.

What is the absolute value of a number?

Its distance from zero.

Which is steeper, a line with gradient −5 or a line with gradient 2?

Gradient −5. Steepness goes by the absolute value, and 5 is further from zero than 2.

Lines with gradients 3 and −3: equally steep? Parallel?

Equally steep, but not parallel. One rises and one falls, so they cross.

What must two lines have to be parallel?

Exactly the same gradient.

Why is counting squares a risky way to find a gradient?

One square is not always a step of one. Read the scales and move a real distance of 1 along the x axis.

When can you read the gradient straight from a table of values?

Only when the x values go up by one each time. Then the gradient is how much y changes from row to row.

A step of one misses the line. Right 3 and up 2 gives what gradient?

Divide both by 3 with a ratio table: a change in x of 1 goes with a change in y of 2/3, so the gradient is 2/3.

Why write a gradient of −4/3 as a fraction, not −1.3?

The fraction is exact. −1.3 is not exactly −4/3.

How can you tell if points A, B and C are on one straight line?

The gradient from A to B must equal the gradient from B to C.

What is the y intercept?

The coordinate where the graph crosses the y axis. It happens when x is zero, so you write it like (0, 2).

How do you find the y intercept from an equation?

Substitute x = 0. For y = x + 4 the y intercept is (0, 4).

A table does not include x = 0. How do you find the y intercept?

Find the gradient, then count back to x = 0 along the line, one gradient per step.

Does a vertical line have a y intercept?

No.

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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Gradient and y-intercept of a linear graph

Tilt the line. Read the step.

Three questions about every line

Where does the working go wrong?

Your turn to supply the steps

Where does y = x + 2 cross the y axis?

More KS3 Maths topics