Straight Line Graphs: y = mx + c
Straight line graphs use the equation y = mx + c, where m is the gradient (how steep the line is, and whether it slopes up or down) and c is the y-intercept (the value of y where the line crosses the y-axis, at x = 0). Any equation written in this form can be plotted by starting at the y-intercept and stepping across using the gradient, and any straight line drawn on a grid can be described by finding its gradient and y-intercept and writing them in this form.
Before you start
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Method
- Compare the equation to y = mx + c: the number multiplying x is the gradient m, and the number added or subtracted on its own is the y-intercept c.
- To plot the line, start by plotting the y-intercept (0, c) on the y-axis.
- From that point, use the gradient as a step: move 1 square right, then m squares up if m is positive, or m squares down if m is negative.
- Repeat this step to find at least two more points, then join all the points with a single straight line, extending it across the grid.
- To find the equation of a line already drawn, read the y-intercept from where it crosses the y-axis, then calculate the gradient as (change in y) / (change in x) between two clear points on the line.
- Remember that a horizontal line has equation y = a number (gradient 0), and a vertical line has equation x = a number (the gradient is undefined).
Worked example
A line has equation y = 2x - 3. (a) State the gradient and the y-intercept. (b) Work out the y-coordinate when x = 4. (c) Find the coordinates of the point where the line crosses the x-axis.
- Compare y = 2x - 3 to y = mx + c: the gradient is m = 2 and the y-intercept is c = -3.
- For (b), substitute x = 4: y = 2(4) - 3 = 8 - 3 = 5, so the point is (4, 5).
- For (c), the line crosses the x-axis where y = 0, so solve 0 = 2x - 3.
- Add 3 to both sides: 2x = 3, then divide by 2: x = 1.5, so the point is (1.5, 0).
Practice questions
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Q1State the gradient and y-intercept of y = 5x + 2.Show answer
Answer: Gradient = 5, y-intercept = 2.
Q2State the gradient and y-intercept of y = -3x + 7.Show answer
Answer: Gradient = -3, y-intercept = 7.
Q3Write down the equation of the line with gradient 4 and y-intercept -1.Show answer
Answer: y = 4x - 1
Q4A line has equation y = x + 6. Work out y when x = -2.Show answer
Answer: y = -2 + 6 = 4
Q5Find the gradient of the line joining the points (0, 1) and (2, 9).Show answer
Answer: Gradient = (9 - 1) / (2 - 0) = 8 / 2 = 4
Q6State the equation of the horizontal line that passes through the point (0, -5).Show answer
Answer: y = -5
Q7Does the point (3, 10) lie on the line y = 3x + 1? Explain your answer.Show answer
Answer: Yes, because substituting x = 3 gives y = 3(3) + 1 = 10, which matches the point's y-coordinate.
Q8Find the coordinates of the point where the line y = 4x - 8 crosses the x-axis.Show answer
Answer: Set y = 0: 0 = 4x - 8, so 4x = 8 and x = 2. The point is (2, 0).
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A line passes through the points (0, 5) and (4, -3). (a) Work out the gradient of the line. (b) Write down the equation of the line.
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The line L has equation y = 3x - 6. (a) Find the coordinates of the point where L crosses the x-axis. (b) Find the coordinates of the point where L crosses the y-axis. (c) A second line, M, is parallel to L and passes through the point (0, 4). Write down the equation of line M.
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Free printable worksheet
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This topic is chapter 31 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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