Other graphs

Apart from straight lines and parabolas, you need to recognise cubic, reciprocal and exponential graphs. Their shapes come from their equations.

Key facts

  • Cubic y = x³: an S-shape through the origin.
  • Reciprocal y = 1/x: two curves (a hyperbola) that never touch the axes.
  • Exponential y = aˣ: passes through (0, 1), shoots up to the right and flattens towards the x-axis to the left.
  • CAPS hyperbola: y = a/(x + p) + q has asymptotes x = −p and y = q.

Worked examples

Example 1

Which graph never crosses either axis: y = x³, y = 1/x or y = 2ˣ?

  1. y = x³ goes through (0, 0).
  2. y = 2ˣ crosses the y-axis at (0, 1).
  3. y = 1/x: x can’t be 0 and y is never 0.

Answer: y = 1/x

Example 2

Find the asymptotes of y = 3/(x − 2) + 1.

  1. The denominator is 0 when x = 2: vertical asymptote x = 2.
  2. The +1 shifts it up: horizontal asymptote y = 1.

Answer: x = 2 and y = 1