Quadratic functions, completing the square and the discriminant

Learning objectives

Express any quadratic in the completed-square form a(x + b)² + c, and read off the maximum or minimum value, the coordinates of the vertex and the equation of the line of symmetry

Sketch the graph of a quadratic showing the vertex, the axis intercepts and the direction in which it opens, using the sign of a

Solve a quadratic equation by factorising, by completing the square and by the quadratic formula, and choose the most efficient method for the given coefficients

Use the discriminant b² − 4ac to determine whether a quadratic has two distinct real roots, a repeated root or no real roots, and apply this condition to find unknown coefficients

Solve a quadratic inequality by finding the critical values and using a sketch or a sign diagram, and express the solution set using inequality or set notation

Use a quadratic function as a model, for example for the path of a projectile or for a maximum area, and interpret the vertex in the context of the problem