0580

Quadratic Graphs

Coordinate Geometry and Graphs

Using the crossings

  • If the graph shows both x-axis crossings, write the equation as y = a(x − x₁)(x − x₂) using those two values
  • Then substitute one further known point, usually the y-axis crossing, and solve for a
  • Expanding at the end is optional unless the question asks for a particular form
  • Give the line of symmetry as an equation, x = p, rather than as a single number
Exam tip

These papers call it the turning point, or the highest or lowest point. They never call it a vertex — the word does appear, but only for the corner of a polygon or the apex of a pyramid. "Parabola" and "the roots" do not appear at all.

Worked example

Building the equation from the crossings

Find the equation of the curve shown.

A u-shaped quadratic curve marked with three points: it crosses the y-axis at (0, 24) and the x-axis at (2, 0) and (3, 0)
Source: Quadratic graphs by Save My Exams

Solution:

  • The two x-axis crossings are at x = 2 and x = 3, so start from y = a(x − 2)(x − 3)
  • Substitute the third point, x = 0 and y = 24
  • 24 = a(0 − 2)(0 − 3)
  • 24 = a × (−2) × (−3), so 24 = 6a and a = 4
  • The equation is y = 4(x − 2)(x − 3)
  • Expanded, that is y = 4x² − 20x + 24
  • Check: substituting x = 0 gives 24, which is the y-axis crossing shown

Using the turning point

  • If the graph shows the turning point instead, write the equation as y = a(x − p)² + q with those coordinates filled in
  • Substitute one other point on the curve and solve for a
Worked example

Building the equation from the turning point

Find the equation of the curve shown.

A u-shaped quadratic curve with its turning point marked at (9, −16) and one further point on the curve marked at (2, 82)
Source: Quadratic graphs by Save My Exams

Solution:

  • The turning point is at (9, −16), so start from y = a(x − 9)² − 16
  • Substitute the other point, x = 2 and y = 82
  • 82 = a(2 − 9)² − 16
  • 82 = a × (−7)² − 16, so 82 = 49a − 16
  • 98 = 49a, giving a = 2
  • The equation is y = 2(x − 9)² − 16
  • Expanded, that is y = 2x² − 36x + 146