0580

Quadratic Graphs

Coordinate Geometry and Graphs

Shape and symmetry

  • A quadratic graph has equation y = ax² + bx + c, with a not zero
  • The curve is smooth and has a vertical line of symmetry, and it is drawn freehand rather than with a ruler
  • A positive a opens the curve upwards, giving a u-shape with a lowest point
  • A negative a opens it downwards, giving an n-shape with a highest point
  • That highest or lowest point is the turning point, and the line of symmetry is the vertical line through it
The two quadratic shapes side by side: with a greater than 0 the curve is a u-shape opening upwards with a minimum point at the bottom, and with a less than 0 it is an n-shape opening downwards with a maximum point at the top
Source: Quadratic graphs by Save My Exams
  • The curve always meets the y-axis, at (0, c), because substituting x = 0 leaves only the constant
  • Against the x-axis there are three possibilities: it cuts through at two points, just touches at one, or misses the axis entirely, depending on where the turning point sits

Where it crosses the x-axis

  • The crossing points are the values of x that make y zero, so they come from solving ax² + bx + c = 0
  • Any of the three quadratic-equation methods will find them: factorising, completing the square, or the formula
  • Two crossings mean the turning point is on the opposite side of the axis from the opening; one crossing means the turning point sits exactly on the axis
  • No crossing means the whole curve stays on one side of the axis, which the completed square form shows immediately
  • The line of symmetry sits half way between two crossings, so averaging them gives its equation quickly