Straight-line graphs: revision notes and practice

What you will learn

• Read coordinates and scales on positive and negative axes.
• Calculate and interpret the gradient of a straight line.
• Find and use an equation in the form y = mx + c.
• Check a point and distinguish exact algebra from a graphical estimate.

Key rules

Gradient

m = change in y / change in x. Keep the direction of both differences consistent.

Line equation

In y = mx + c, m is the gradient and c is the y-intercept.

Check a point

Substitute both coordinates into the line equation. The equality must hold.

Worked examples

Example 1: Find the equation through (−1, 1) and (2, 7).

1. Gradient = (7 − 1)/(2 − (−1)) = 6/3 = 2.
2. Use (−1, 1): 1 = 2(−1) + c, so c = 3.

y = 2x + 3

Example 2: Does (3, 8) lie on y = 2x + 3?

1. At x = 3, the line gives y = 2 × 3 + 3 = 9.
2. The given point has y = 8.

No; the coordinates do not satisfy the equation.

A practice question

Line A passes through (-4, -9) and (4, -25), as shown on the graph. Calculate the gradient of Line A.

1. Understand what a coordinate means

A coordinate pair records two linked values in a fixed order. In (−2, 5), x is −2 and y is 5. Move horizontally to the x-position first, then vertically to the y-position. The negative x-value places the point to the left of the y-axis. Reading the pair backwards describes a different point, so label axes and keep the order clear throughout a calculation.

A graph shows many possible coordinate pairs, not just the points used to draw it. If a point lies on a line, its two coordinates satisfy that line’s equation. A straight line continues between and beyond its marked points unless a context restricts its domain. Two plotted points are enough to determine a non-vertical straight line, but using additional points is a useful check.

2. Read the scale before reading the line

Count the value represented by one interval on each axis. A horizontal square might represent one unit while a vertical square represents five units. Counting squares alone then gives a misleading gradient. Use coordinate values to calculate actual changes. Also check whether zero is visible and whether the axes include negative values; a crossing below the origin has a negative y-intercept.

Read a plotted point at the centre of its marker, rather than at the edge of a thick line. A graph drawn on a screen may support an estimate, but the stated coordinate values are exact givens. If an exercise supplies both coordinates and a picture, use the exact coordinates for the calculation and the picture to check whether your answer has a sensible sign and size.

3. Calculate gradient as a ratio

Gradient measures how much y changes for each one-unit increase in x. For points (−1, 1) and (2, 7), the horizontal change is 2 − (−1) = 3 and the vertical change is 7 − 1 = 6. The gradient is therefore 6/3 = 2. The line rises two y-units for every x-unit moved to the right.

Subtract the coordinates in the same order in the numerator and denominator. Reversing both differences leaves the ratio unchanged; reversing only one gives the wrong sign. Choose points well apart to reduce reading error on a drawn graph. You do not need to divide the coordinates of one point: that method only works in the special case of a line through the origin.

m = (y₂ − y₁)/(x₂ − x₁)

4. Interpret positive, negative and zero gradients

A positive gradient means the line rises as x increases. A negative gradient means it falls. For example, moving from (0, 4) to (3, −2) changes y by −6 and x by 3, so the gradient is −2. The negative sign describes direction; it does not mean that every coordinate on the line must be negative.

A horizontal line has zero gradient because its y-value never changes. A vertical line cannot be written as y = mx + c with a finite gradient: its horizontal change is zero, so the usual ratio would require division by zero. This app’s line questions use non-vertical lines. Equal gradients describe parallel straight lines, provided their intercepts differ; equal gradients and equal intercepts describe the same line.

5. Connect gradient and intercept to an equation

In y = mx + c, m is the gradient and c is the y-intercept. The intercept is the y-value when x = 0, because mx then equals zero. The equation y = 2x + 3 has gradient 2 and crosses the y-axis at (0, 3). The intercept is not the point where the line crosses the x-axis.

If the y-axis crossing is difficult to read, use a known point after calculating the gradient. For the line through (−1, 1) and (2, 7), m = 2. Substitute the first point: 1 = 2(−1) + c. This gives c = 3, so the equation is y = 2x + 3. Substitute the other point to check that the same equation gives 7.

y = mx + c

6. Find missing coordinates using the equation

To find y for a given x, replace x with the entire number, including its sign. On y = −3x + 4, when x = −2, y = −3(−2) + 4 = 10. The two negative signs multiply to a positive value. Keep the intercept as a separate addition after the multiplication; do not multiply it by x.

To find x for a given y, solve the resulting linear equation. If y = −8 on the same line, write −8 = −3x + 4. Subtract 4 to get −12 = −3x, then divide by −3 to obtain x = 4. Check by substitution: −3(4) + 4 = −8. The graph provides a visual check, while the algebra gives the exact coordinate.

7. Decide whether a point belongs to a line

Substitute the point’s x-value into the equation and compare the result with its stated y-value. The point (3, 8) is not on y = 2x + 3, since the equation gives 9 when x = 3. Write the calculation and the comparison, rather than relying on a point appearing close to a thick plotted line.

For a line described by two points rather than an equation, first calculate the gradient and intercept. A third point belongs to the line only if it follows the same relationship. One matching coordinate is not enough. A point with the correct x-value but the wrong y-value is a different position, and a point on the visible axis may still be unrelated to the required line.

8. Present a complete graph solution

Start by recording two exact coordinate pairs and identifying what the question asks. A gradient question needs a ratio with both differences visible. An equation question needs both the gradient and the intercept, followed by an equation with y isolated. A coordinate question needs the requested value, not just a substituted expression. Include contextual units when the axes represent quantities such as time and cost.

Finally, use an independent check. Compare the sign of your gradient with the direction of the line, substitute a known point into your equation, or check the result in the original equation after rearranging. These checks catch reversed subtraction, misread scales and lost negative signs. The original practice here covers selected line skills; it does not cover every GCSE graph type or every exam-board application.

Check your understanding

How to earn the marks

• Read the axis intervals before choosing coordinates.
• Use the same subtraction order for both changes.
• For a line equation, show both m and c before writing y = mx + c.
• Use exact stated coordinates, then check the result against the graph.

A common mistake

Gradient is a ratio of coordinate changes, not the y-coordinate divided by the x-coordinate of one point.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Mathematics — A8, A9, A10

  Straight-line gradients, intercepts and equations are available at both tiers. Perpendicular gradients and non-linear graph extensions are not included in this topic bank.

• Edexcel GCSE Mathematics — A8, A9, A10

  Straight-line gradients, intercepts and equations are available at both tiers. Perpendicular gradients and non-linear graph extensions are not included in this topic bank.

• OCR GCSE Mathematics — 7.01, 7.02

  Straight-line gradients, intercepts and equations are available at both tiers. Perpendicular gradients and non-linear graph extensions are not included in this topic bank.

Keep your next step small

Try five questions, read the feedback, and revisit one thing you found difficult. The app tracks a 20-answer practice goal separately from accuracy.

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