Gradient and intercepts Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Gradient and intercepts questions. See exactly how to solve problems on reading a graph, y-intercept, gradient, rise over run.

reading a graphy-interceptgradientrise over runy = mx + cgradient between two points
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
The straight line shown on the graph crosses the yy-axis. Write down the coordinates of the point where the line crosses the yy-axis.

Worked solution

  1. Find the y-axis

    x=0x = 0

    The y-axis is the vertical line where x=0x = 0.

  2. Read where the line crosses

    y=2 when x=0y = 2 \text{ when } x = 0

    Follow the line to where it cuts the y-axis; it passes through a height of 2.

  3. Write the coordinates

    (0,2)(0, 2)

    The crossing point is written as a coordinate: x=0x = 0, y=2y = 2.

Answer
(0,2)(0, 2)
Question 2
1 markeasy
The graph shows a straight line passing through the origin. Use the graph to write down the gradient of the line.

Worked solution

  1. Pick two points on the line

    (0,0) and (1,2)(0, 0) \text{ and } (1, 2)

    Choose points where the line passes through grid corners.

  2. Work out the rise and the run

    rise=2,run=1\text{rise} = 2, \quad \text{run} = 1

    Going from (0, 0) to (1, 2) the line goes up 2 and across 1.

  3. Gradient = rise ÷\div run

    gradient=21=2\text{gradient} = \frac{2}{1} = 2

    The line rises 2 units for every 1 unit across, so the gradient is 2.

Answer
22
Question 3
1 markeasy
A straight line has equation y=2x−1y = 2x - 1. Write down the yy-intercept.

Worked solution

  1. Recall the straight-line form

    y=mx+cy = mx + c

    In this form c is the y-intercept.

  2. Compare with the equation

    y=2x+(−1)y = 2x + (-1)

    Here m=2m = 2 and c=−1c = -1.

  3. State the y-intercept

    c=−1c = -1

    The line crosses the y-axis at -1.

Answer
−1-1
Question 4
2 markseasy
Work out the gradient of the line joining the points (0,0)(0, 0) and (2,6)(2, 6).

Worked solution

  1. Write the gradient formula

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

    Gradient is the change in y divided by the change in x.

  2. Substitute the coordinates

    m=6−02−0m = \frac{6 - 0}{2 - 0}

    Subtract the y-values and the x-values in the same order.

  3. Simplify

    m=62=3m = \frac{6}{2} = 3

    The gradient is 3.

Answer
33
Question 5
2 markseasy
Work out the gradient of the line joining the points (1,2)(1, 2) and (3,8)(3, 8).

Worked solution

  1. Write the gradient formula

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

    Gradient is the change in y divided by the change in x.

  2. Substitute the coordinates

    m=8−23−1m = \frac{8 - 2}{3 - 1}

    Take the differences in the y-values and the x-values.

  3. Simplify

    m=62=3m = \frac{6}{2} = 3

    The gradient is 3.

Answer
33

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