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Straight Line Graphs: y = mx + c

Understand straight line graphs — what gradient and y-intercept mean, how to plot them, how to find equations, and real-life applications like distance-time graphs.

Every straight line on a graph can be written as y = mx + c. Once you understand what m and c mean, you can read, draw and create any straight line graph.

What Do m and c Mean?

y = mx + c
  • m = gradient (how steep the line is — rise ÷ run)
  • c = y-intercept (where the line crosses the y-axis)

Think of it like walking up a hill. m is how steep the hill is.c is where you started (your altitude at the beginning).

Finding the Gradient

Gradient = Change in y ÷ Change in x = (y₂ - y₁) ÷ (x₂ - x₁)

Pick any two points on the line, find the difference in their y-values and divide by the difference in their x-values.

Worked Example 1 — Finding Gradient and Equation

A line passes through (1, 3) and (4, 9). Find its equation.

  1. Gradient m = (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2
  2. Use y = mx + c with point (1, 3): 3 = 2(1) + c → c = 1
  3. Equation: y = 2x + 1

Plotting from an Equation

Worked Example 2 — Plotting y = 3x − 2

  1. The y-intercept is −2, so plot the point (0, −2)
  2. The gradient is 3, so from that point go right 1, up 3 to get (1, 1)
  3. Plot another: go right 1 again, up 3 → (2, 4)
  4. Draw a straight line through all three points

Parallel & Perpendicular Lines

  • Parallel lines have the same gradient. y = 2x + 1 and y = 2x + 5 are parallel.
  • Perpendicular lines have gradients that are negative reciprocals. If one has gradient 2, the perpendicular has gradient −1/2.

Worked Example 3

Find the equation of the line perpendicular to y = 3x + 1 that passes through (6, 2).

  1. Original gradient = 3
  2. Perpendicular gradient = −1/3
  3. y = −1/3 x + c, substitute (6, 2): 2 = −1/3(6) + c → 2 = −2 + c → c = 4
  4. Answer: y = −1/3 x + 4

Real-Life Graphs

Distance-time graphs: gradient = speed. A steeper line means moving faster. A flat line means stopped.

Conversion graphs: convert between units like °C and °F, or miles and kilometres.

Try It Yourself

  1. Find the gradient of a line through (2, 5) and (6, 13). (Answer: 2)
  2. What is the y-intercept of y = −4x + 7? (Answer: 7)
  3. A line is parallel to y = 5x − 3 and passes through (0, 2). What's its equation? (Answer: y = 5x + 2)