Drawing straight-line graphs from an equation
Build a table, plot accurate points and draw the complete line with a ruler.
Section Paper 4 · Extended · Calculator
Core and Extended. Core equations are given as y = mx + c unless a table is supplied; rearranging other forms is taught later as Extended content. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Substitute positive and negative values into an expression.
- Plot an ordered pair on a labelled grid.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is −2(−1) + 3?
What you will learn
- Generate coordinate pairs from y = mx + c.
- Draw and check a straight-line graph on a suitable scale.
- Recognise horizontal and vertical lines.
An equation describes every point on the line
For y = 2x − 1, choosing x = −1 gives y = −3, so (−1, −3) is one point on the graph. Choosing x = 0 gives (0, −1), and x = 2 gives (2, 3). Each pair satisfies the same equation. A table records the input x and its corresponding output y; do not mix a y-value with an x-value from another column.
Work through each substitution in brackets, especially for negative inputs. For y = −2x + 3 and x = −1, y = −2(−1) + 3 = 5. The minus sign is part of the multiplier. Choose x-values that give points spread across the available grid, rather than three points crowded together.
Plot, check, then join
Label both axes and choose a scale that fits the required values. Plot at least three points, although two distinct points determine a line. The third is a useful check: if it does not lie on the line through the first two, revisit the arithmetic, scales and plotting before drawing a final line.
Use a ruler to draw a straight line across the requested graph interval. Do not join points with a curved stroke or stop the graph at the first and last calculated point unless the question restricts the domain. A line's intercept at x = 0 is especially useful for checking the constant c in y = mx + c.
Horizontal and vertical equations
For y = 3, every point has vertical coordinate three, while x is free to vary. The graph is horizontal. For x = −2, every point has horizontal coordinate negative two, while y varies, so the graph is vertical. A vertical line cannot be written as y = mx + c with a finite gradient.
The equations x = 0 and y = 0 describe the y-axis and x-axis respectively. Name a line by the coordinate that stays fixed. A value table supplied with a question can also be plotted directly: use the given pairs, check the scale and decide whether the points lie on a straight line.
Pause and explain
Which point lies on y = 2x − 1?
Worked example
Draw y = −2x + 3 for −1 ≤ x ≤ 3.
Show the worked solution
- For x = −1, 0, 1, 2, 3, calculate y = 5, 3, 1, −1, −3.
- Plot the five pairs on labelled axes, using one consistent scale on each axis.
- Check that (0, 3) is the y-intercept and all points align; draw the straight segment over the stated interval.
Answer The line passes through (−1, 5), (0, 3), (1, 1), (2, −1), (3, −3).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Make a table and draw y = 3x + 1 for x = −1, 0, 1, 2.
Give me a hint
Find each y-value before plotting the pair.
Compare my reasoning
- The y-values are −2, 1, 4 and 7.
- Plot (−1, −2), (0, 1), (1, 4), (2, 7).
- The points align and cross the y-axis at one; draw the line with a ruler.
Pairs: (−1, −2), (0, 1), (1, 4), (2, 7)
Look for these in your work
- Computed every substitution correctly.
- Used a straight line and labelled axes.
Draw y = 4 and x = −1 on the same axes. State their intersection.
Give me a hint
Identify which coordinate each equation holds constant.
Compare my reasoning
- Draw a horizontal line four units above the origin.
- Draw a vertical line one unit left of the origin.
- Their intersection has x = −1 and y = 4.
Intersection (−1, 4)
Look for these in your work
- Distinguished horizontal and vertical equations.
- Read the intersection as an ordered pair.
A tank initially holds 6 litres and gains 2 litres each minute. Draw V = 2t + 6 for 0 ≤ t ≤ 4 and explain why the negative-time part is excluded.
Give me a hint
Use time on the horizontal axis and volume on the vertical axis.
Compare my reasoning
- For t = 0, 1, 2, 3, 4, volumes are 6, 8, 10, 12, 14 litres.
- Plot the pairs and join them over the interval from zero to four minutes.
- Negative t is before the modelled filling starts, so that part is outside the stated situation.
Line segment from (0, 6) to (4, 14)
Look for these in your work
- Labelled quantities and units.
- Respected the context's restricted time interval.
Common mistakes
- Multiplying negative inputs without brackets.
- Drawing a curve through linear points.
- Confusing x = k with a horizontal line.
What can you explain now?
Give the y-intercept of y = −3x − 2.
Compare with the explanation
(0, −2)
At the y-axis x = 0, so substitution leaves y = −2.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Draw a new line from a table of three values tomorrow, then check a fourth point by substitution.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.