Quadratic Graphs | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Definition of Quadratic Function

• A quadratic function has the form y = ax² + bx + c where a ≠ 0.
• Quadratic functions are common in mathematics, making it important to understand their key features.

What does a quadratic graph look like?

• The shape formed by a quadratic graph is called a parabola.
• A parabola can appear as either a "u-shape" or an "n-shape".
  • If the coefficient of x² is positive, the parabola is a "u-shape".
  • If the coefficient of x² is negative, the parabola is an "n-shape".
• A quadratic graph always intersects the y-axis.
• A quadratic graph may intersect the x-axis twice, once, or not at all.
  • The points where the graph intersects the x-axis are known as the roots.
• A "u-shape" parabola has a minimum point (the bottom of the "u").
• An "n-shape" parabola has a maximum point (the top of the "n").
• Minimum and maximum points are types of turning points.

How to Sketch a Quadratic Graph?

• We can create a table of values for the function and plot it accurately, but often a sketch highlighting key features is sufficient.
• The most important features of a quadratic are:
  • Its overall shape: a "u-shape" or an "n-shape".
  • Its y-intercept.
  • Its x-intercept(s), also known as the roots.
  • Its minimum or maximum point (turning point).
• If the coefficient a in ax² + bx + c is positive, the quadratic will be a "u-shape".
• If the coefficient a in ax² + bx + c is negative, the quadratic will be an "n-shape".
• The y-intercept of y = ax² + bx + c is (0, c).
• The roots, or x-intercepts, are the solutions to y = 0, i.e., ax² + bx + c = 0.
• You can solve a quadratic by factorizing, completing the square, or using the quadratic formula.
  • There may be 2, 1, or 0 solutions, and therefore 2, 1, or 0 roots.
• The minimum or maximum point of a quadratic can be found by:
  • Completing the square: Once the quadratic is written in the form y = a(x − q)² + r, the minimum or maximum point is given by (q, r).
    • Be careful with the sign of the x-coordinate. For example, if the equation is y = (x − 3)² + 2, the minimum point is (3, 2), but if the equation is y = (x + 3)² + 2, the minimum point is (−3, 2).
  • Using differentiation: Solving dx/dy = 0 will find the x-coordinate of the minimum or maximum point. You can then substitute this value into the quadratic equation to find the y-coordinate.