Quadratic Functions

You will be able to

• Express ax² + bx + c in the form a(x + p)² + q by completing the square, showing each step.
• Deduce the maximum or minimum value of a quadratic and the value of x at which it occurs from the completed-square form.
• State the range of a quadratic function for all real x, as f(x) ≥ k or f(x) ≤ k.

You will be able to

• Substitute a linear equation into a quadratic one to form a single quadratic equation in x.
• Use the discriminant to decide whether a line meets a curve at two points, touches it (a tangent) or does not meet it.
• Find the value, or the range of values, of an unknown constant k or m for which a line is a tangent to, cuts, or does not meet a curve.

You will be able to

• Find the critical values of a quadratic inequality by solving the corresponding equation.
• Solve a quadratic inequality using a sketch or a sign diagram, choosing the inside or the outside region.
• Write the solution set correctly, e.g. x < −1 or x > 3, or −1 ≤ x ≤ 3, using strict or inclusive inequalities as the question requires.