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Straight-line graphs

Interpret and construct linear graphs from equations, points and real-world rates.

12 free questions · Notes · Worked solutions

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What you will practise

  • Use y=mx+c and identify gradient and intercept.
  • Calculate gradient from two coordinates.
  • Find intercepts and solve simple graphical relationships.
  • Interpret line graphs as proportional or affine relationships.

Revision notes

Gradient and intercept

A straight line can be written y=mx+c. The gradient m is change in y divided by change in x; c is the y-intercept. A positive gradient rises from left to right.

Plotting and reading

A table of values gives coordinate pairs (x,y). Plot coordinates using the same scale on each axis, then draw a straight line through them. The x-intercept occurs where y=0.

Parallel lines

Parallel straight lines have equal gradients. A line through a point can be described by its gradient and one known coordinate, then rearranged to y=mx+c.

Common checks

Gradient is not the angle of the line. Keep the coordinate order consistent in Δy/Δx. A line through the origin has c=0 and represents direct proportion.

A worked example

A line passes through A(2,5) and B(6,13). Find its equation in the form y=mx+c.

  1. Gradient m=(13−5)/(6−2)=2.
  2. Use y=2x+c and substitute (2,5).
  3. 5=4+c, so c=1.
  4. The equation is y=2x+1.

y=2x+1

Try it yourself

These are original practice questions, not official past-paper questions. Practice points count correct final answers; working is not automatically assessed.

Choose a question to begin.
Question 1 · Standard1 practice point

For y=−3x+4, what is the gradient?

Need a hint?

In y=mx+c, the coefficient of x is m.

Worked solution
  1. Compare with y=mx+c.
  2. The coefficient of x is −3.
Question 2 · Standard1 practice point

Which equation has gradient 2 and y-intercept −5?

Need a hint?

Place the gradient beside x and the intercept as the constant.

Worked solution
  1. Use y=mx+c with m=2 and c=−5.
  2. This gives y=2x−5.
Question 3 · Standard1 practice point

A line passes through (0,−2) and (4,6). What is its gradient?

Need a hint?

Use (change in y)/(change in x).

Worked solution
  1. Δy=6−(−2)=8.
  2. Δx=4−0=4; gradient=8÷4=2.
Question 4 · Standard1 practice point

Which point lies on y=3x−1?

Need a hint?

Substitute each x-coordinate and compare the resulting y.

Worked solution
  1. For x=3, y=3(3)−1=8.
  2. Thus (3,8) lies on the line.
Question 5 · Foundation1 practice point

A line is y=4x−7. Find y when x=3.

Need a hint?

Substitute x=3.

Worked solution
  1. y=4(3)−7.
  2. y=12−7=5.
Question 6 · Standard2 practice points

Calculate the gradient of the line through (−1,2) and (3,10).

Need a hint?

Use the same point order for both differences.

Worked solution
  1. Δy=10−2=8.
  2. Δx=3−(−1)=4.
  3. m=8/4=2.
Question 7 · Foundation1 practice point

Find the y-intercept of y=−2x+9.

Need a hint?

Set x=0.

Worked solution
  1. At the y-intercept, x=0.
  2. y=−2(0)+9=9.
Question 8 · Standard2 practice points

Find the x-intercept of y=5x+15.

Need a hint?

At the x-intercept, y=0.

Worked solution
  1. Set 0=5x+15.
  2. 5x=−15, so x=−3.
Question 9 · Standard2 practice points

A taxi fare y pounds is given by y=2.5x+4, where x is distance in kilometres. Find the fare for 8 km.

Need a hint?

The fixed fee is added to the distance charge.

Worked solution
  1. Substitute x=8: y=2.5×8+4.
  2. y=20+4=24.
Question 10 · Foundation2 practice points

A straight line passes through (0,3) and (5,18). Find its gradient.

Need a hint?

Calculate rise divided by run.

Worked solution
  1. Δy=18−3=15.
  2. Δx=5−0=5.
  3. Gradient=15/5=3.
Question 11 · Standard2 practice points

A line has gradient −4 and passes through (2,1). Find c in y=−4x+c.

Need a hint?

Substitute the known coordinate.

Worked solution
  1. 1=−4(2)+c.
  2. 1=−8+c, so c=9.
Question 12 · Stretch3 practice points

A line is y=2x+1. A second line is parallel and passes through (0,−4). Find the second line's y-value when x=6.

Need a hint?

Parallel lines share the gradient; use the given point to find the intercept.

Worked solution
  1. The second line has gradient 2.
  2. At x=0 its y-intercept is −4, so y=2x−4.
  3. At x=6, y=12−4=8.

Coverage and sources

Selected IGCSE concepts, with a mix of introductory and more demanding questions. Check your own specification. AI drafted; independent teacher review is pending. Editorial details.

Cambridge IGCSE Mathematics 0580 syllabus

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