GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Gradients and intercepts of linear functions

Every straight line is pinned down by two numbers: how steep it is, and where it crosses the y-axis. Spot them and y = 7 − 4x becomes a picture.

y = −2x + c: lift the line. What changes, and what doesn't?

Constant c 0

Constant c: 0. Gradient: -2

The dot you're holding is where the line crosses the y-axis, at (0, c). Lift it from 0 to 4: the whole line slides up, and the crossing point moves from (0, 0) to (0, 4). Now look at the gradient. It stays at −2 the whole time, because every one of these lines still drops 2 for each 1 you move right. That's what a gradient is: the change in y for each 1 across. So y = −2x + 4 is just y = −2x lifted 4 units.

GRADIENT FROM A GRAPH

Build the gradient triangle, one step at a time

A straight line passes through the grid points (0, 2) and (3, −2). Find its gradient.

Pick two points where the line crosses grid lines exactly: (0, 2) and (3, −2).Any two points on the line give the same gradient. Grid points are just the easiest to read accurately.
Draw the triangle moving in the positive x direction, from (0, 2) across to x = 3. Change in x = 3 − 0 = 3.
missing step

Which line is step 3?

Now go down from y = 2 to y = −2. Change in y = 2 − (−2) = 4.Now go down from y = 2 to y = −2. Change in y = −2.Now go down from y = 2 to y = −2. Change in y = −2 − 2 = −4.

SPOT THE SLIP

The triangle's right. The answer isn't.

On this graph the x-axis is numbered 0, 2, 4, 6, 8, so each square across is worth 2. The y-axis is numbered 0, 1, 2, 3, 4, 5, so each square up is worth 1. A straight line passes through the grid points (2, 5) and (8, 1). Find its gradient.

A student's answer — which line goes wrong?

I picked two grid points on the line: (2, 5) and (8, 1).
I drew the gradient triangle from the left point, moving right.
It goes 3 squares across, so the change in x is 3.
It goes 4 squares down, so the change in y is −4.
Gradient = −4 ÷ 3 = −4/3.

READING AN EQUATION

Which number is the gradient?

Here's the equation of a straight line: y = 7 − 4x

What are its gradient and y-intercept? Pick the one closest to what you think right now.

Gradient 7 and y-intercept (0, −4). The first number is the gradient and the second is the intercept.Gradient −4 and y-intercept (0, 7), because the gradient is whatever multiplies x.Gradient 4 and y-intercept (0, 7). The minus is just a take-away sign.Gradient −4, and the y-intercept is just 7, because the intercept is the number on its own.

How sure are you?

I'm sureI think soI'm guessing

Gradient 7 and y-intercept (0, −4). The first number is the gradient and the second is the intercept.

This is a common misconception

Reading by position only works when the equation is written in the order y = mx + c, and this one isn't. The gradient is the number multiplying x, which here is −4. Reorder it to y = −4x + 7 and the constant, 7, gives the y-intercept (0, 7).

Gradient −4 and y-intercept (0, 7), because the gradient is whatever multiplies x.

This matches the evidence

Yes. Reorder it to y = −4x + 7: the coefficient of x, sign included, is the gradient, and the constant gives the y-intercept (0, 7).

Gradient 4 and y-intercept (0, 7). The minus is just a take-away sign.

This is a common misconception

The minus sign belongs to the 4x term: y = 7 − 4x is the same as y = −4x + 7. So the coefficient of x is −4, and y goes down 4 for every 1 across. Drop the sign and you'd have a line sloping the wrong way.

Gradient −4, and the y-intercept is just 7, because the intercept is the number on its own.

This is a common misconception

The gradient is spot on. But an intercept is a point, so give it as a coordinate. 7 is the y-coordinate of the y-intercept; the intercept itself is (0, 7), because x is 0 on the y-axis.

Don't read m and c by their position. m is the number multiplying x, sign and all, and c is the number on its own. Reorder to y = mx + c and both are easy to spot.

Maths · Algebra

When y isn't the subject, rearrange first

Step through each line. The label on the right says what was done to both sides.

GoalFind the gradient and y-intercept of 4x + 5y = 11

4x + 5y = 11

y isn't the subject yet, so the gradient is hidden. It is not 4.

Step 1 of 5

y isn't the subject yet, so the gradient is hidden. It is not 4.

From graph to equation, then check it

Problem

From its graph, a straight line has gradient −8/3 and crosses the y-axis at (0, 10). The points (3, 2) and (6, −6) are on the line. Write its equation in the form y = mx + c, rewrite it without fractions, then check both versions.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

In y = mx + c, one number decides how steep the line is and the other decides where it crosses the y-axis. Grab the line below and find out which is which.

What you need to know

The gradient measures how steep a line is: the change in y for a change of 1 in the positive x direction. Uphill from left to right is positive, downhill is negative, and a horizontal line has gradient 0.
To find a gradient from a graph, draw a triangle between two grid points, moving left to right, then work out change in y ÷ change in x. It doesn't matter which two points you use.
Read the changes from the numbers on the axes. A square across doesn't have to be worth the same as a square up.
In y = mx + c, the coefficient of x (m) is the gradient and c is the y-coordinate of the y-intercept, so the line crosses the y-axis at (0, c).
Terms can come in any order: find the coefficient of x, sign included. If y isn't the subject, as in 4x + 5y = 11, rearrange to make it the subject first.
An intercept is a point where a line meets an axis. On the y-axis x = 0, and on the x-axis y = 0.
To write a line's equation from its graph, find m and c and put them into y = mx + c. Check it by substituting a point on the line.

The big picture

Every straight line y = mx + c carries two numbers with two separate jobs. m, the coefficient of x, is the gradient: how much y changes for each 1 you move in the positive x direction. c gives the y-intercept, the point (0, c) where the line crosses the y-axis. You find a gradient from a graph with a triangle and the axis scales, you read m and c from an equation once y is the subject, and you write the equation of a line from its graph and check it by substituting a point.

Key points

1y = mx + c: m is the gradient; (0, c) is the y-intercept.

2Gradient = change in y ÷ change in x, with the triangle drawn left to right.

3Check the axis scales before you count a single square.

4Changing c slides the line up or down. The gradient stays exactly the same.

5Any point on a line satisfies its equation, so substituting a point is your check.

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Worked example

Problem

A line has equation 2y + 6x = 8. Find its gradient and y-intercept, then decide whether the point (3, −5) lies on the line.

⚠ Watch out

Reading m and c by position instead of by role. In y = 7 − 4x the gradient is −4, the coefficient of x, not 7, and the y-intercept is (0, 7). In 4x + 5y = 11 the gradient isn't 4 either: make y the subject first.

🧠

Memory hook

m for mountain: how steep the climb is. c for crossing: where the line crosses the y-axis, at (0, c). Change c and the line just slides, so y = −2x + 4 is y = −2x lifted 4 units.

✓

Check yourself

Without looking back: find the gradient and y-intercept of 3y − 6x = 9. Then explain why counting squares on a graph can give you the wrong gradient.

Flashcards

(13)

What does the gradient of a straight line measure?

How steep it is: the change in y for a change of 1 in the positive x direction.

Positive, negative or zero gradient: how can you tell from the graph?

Uphill from left to right is positive, downhill is negative, and a horizontal line has gradient 0.

How do you find a gradient from a graph?

Draw a triangle between two grid points, moving left to right, then do change in y ÷ change in x.

Does it matter which two points on a line you use for the gradient triangle?

No. Any two points on a straight line give the same gradient.

Why must you check the axis scales before finding a gradient?

A square across may not be worth the same as a square up. From (2, 0) to (4, 10) the gradient is 10 ÷ 2 = 5, whatever the squares look like.

In y = mx + c, what is m?

The coefficient of x: the gradient of the line.

In y = mx + c, what does c tell you?

c is the y-coordinate of the y-intercept, so the line crosses the y-axis at (0, c).

What are the gradient and y-intercept of y = 5x − 7?

Gradient 5, y-intercept (0, −7). In its table of values, each step of +1 in x gives a step of +5 in y.

What is true about the coordinates of any x-intercept?

Its y-coordinate is 0, so substitute y = 0 to find it. For a y-intercept, substitute x = 0.

What are the gradient and y-intercept of 7x + 6y = 5?

Make y the subject: y = 5/6 − (7/6)x. Gradient −7/6, y-intercept (0, 5/6).

What happens to a line y = mx + c when you change c?

It slides up or down (a translation), so its y-intercept moves but its gradient stays the same.

A line has gradient 3 and y-intercept (0, −5). What is its equation?

y = 3x − 5. Rearranged, 3x − y = 5 is the same line.

How do you check an equation you've found from a graph?

Substitute the coordinates of a point on the line. If they satisfy the equation, it checks out.

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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y = −2x + c: lift the line. What changes, and what doesn't?

Build the gradient triangle, one step at a time

The triangle's right. The answer isn't.

Which number is the gradient?

When y isn't the subject, rearrange first

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