Quick questions on Quadratic functions and their graphs explained: O-Level E-Maths

The graph of $y = ax^2 + bx + c$ is a parabola, a smooth symmetric U-shaped curve. If $a > 0$ the parabola opens upward and has a minimum point; if $a < 0$ it opens downward and has a maximum point. The larger $|a|$, the narrower the curve.

The turning point (vertex) is the lowest point of an upward parabola or the highest point of a downward one. Completing the square to write the function as $a(x - h)^2 + k$ shows the turning point directly at $(h, k)$. The minimum or maximum value of $y$ is $k$.

A parabola is symmetric about a vertical line through its turning point, with equation $x = h$. This line of symmetry sits exactly halfway between the two $x$-intercepts when they exist, which is a quick way to find $h$.

When the $x$-intercepts are known, there is a quicker route to the turning point than completing the square: because a parabola is symmetric, the line of symmetry sits exactly midway between the two roots. Average the roots to get the $x$-coordinate of the vertex, then substitute that value into the function to find the minimum or maximum $y$. For $y = x^2 - 2x - 3$ with roots $-1$ and $3$, the axis is $x = \tfrac{-1 + 3}{2} = 1$, and substituting gives $y = -4$, so the vertex is $(1, -4)$. Using symmetry of the roots is the fastest method whenever the quadratic factorises.

The number of times the parabola crosses the $x$-axis matches the discriminant $b^2 - 4ac$ of the quadratic. Two crossings mean a positive discriminant, the curve just touching the axis at its vertex means a zero discriminant (a repeated root), and the curve missing the axis entirely means a negative discriminant with no real roots. So a parabola whose vertex sits above the $x$-axis while opening upward has no real roots. Linking the picture to the discriminant lets you predict, before solving, how many $x$-intercepts to expect and serves as a check on your algebra.

State whether $y = -2x^2 + 3x + 1$ has a maximum or minimum, and why. [1 mark]

Find the $y$-intercept of $y = x^2 + 5x - 6$. [1 mark]

Find the $x$-intercepts of $y = x^2 - x - 12$. [2 marks]