Quadratic functions and parabolas

At x = 0, all terms containing x vanish, so the y-intercept value is the constant term. In a linear cost formula it is the fixed fee; total cost = fixed fee + unit price × quantity. For a quadratic, the intercept is (0, c).

The vertex is where a parabola changes from increasing to decreasing or vice versa. Read its x and y from the graph. A downward-opening parabola has its maximum y at the vertex, representing the ball maximum height and when it occurs.

The vertex is where a parabola changes from increasing to decreasing or vice versa. Read its x and y from the graph. A downward-opening parabola has its maximum y at the vertex, representing the ball maximum height and when it occurs.

For a parabola written as y = (x − m)² + n, the vertex is V(m, n).

The axis of symmetry is the vertical line passing through the vertex, so its equation is:

x = m.

The value n gives the height of the vertex but does not change the equation of the symmetry axis.

The vertex is where a parabola changes from increasing to decreasing or vice versa. Read its x and y from the graph. A downward-opening parabola has its maximum y at the vertex, representing the ball maximum height and when it occurs.

The vertex form of a parabola is y = a(x − m)² + n.

In this form, the vertex is V(m, n). Coefficient a does not change the vertex coordinates; it determines the opening direction and width.

Do not reverse the sign mentally: if the brackets contain x − (−m), the x-coordinate of the vertex is negative.

The zeros of a function are the x-values for which y = 0.

When the function is already factorised, use the zero-product rule:

(x − x₁)(x − x₂) = 0.

A product equals zero when at least one factor equals zero. Therefore solve x − x₁ = 0 and x − x₂ = 0 separately.

The graph of y = ax² is a parabola. The value |a| shows how strongly the parabola is horizontally compressed or stretched.

• The larger |a| is, the narrower the parabola becomes.
• The closer |a| is to zero, the wider the parabola becomes.

The sign of a determines its direction: when a > 0, the parabola opens upwards; when a < 0, it opens downwards.

The sign of coefficient a determines the direction in which the parabola y = ax² opens:

• when a > 0, it opens upwards;
• when a < 0, it opens downwards.

The size of the coefficient changes the width of the parabola, but only its sign determines the opening direction.

At x = 0, all terms containing x vanish, so the y-intercept value is the constant term. In a linear cost formula it is the fixed fee; total cost = fixed fee + unit price × quantity. For a quadratic, the intercept is (0, c).