Read the peak and the variations

The vertex of the parabola

In f(x) = a(x - alpha)^2 + beta, the point S(alpha; beta) is the vertex of the parabola representing f. The vertical line with equation x = alpha is the axis of symmetry of the curve.

Direction of variation

The sign of a determines the shape of the parabola and therefore the variations in f:

Sign of a	Shape	Variations	Extremum
a > 0	upwards	decreasing then increasing	minimum at beta when x = alpha
a < 0	downwards	increasing then decreasing	maximum at beta when x = alpha

Why it works

Since (x - alpha)² ≥ 0 for all x, the term a(x - alpha)² always has the same sign as a (or is equal to 0 if x = alpha). Therefore, f(x) is always greater than or equal to beta if a > 0, and always less than or equal to beta if a < 0.

Example

For f(x) = 2(x - 2)^2 - 3 (a = 2 > 0): the vertex is S(2; -3), f has a minimum of -3 reached at x = 2, f is decreasing on ]∞; 2] and then increasing on [2; +∞[.

Common pitfall

Be careful with the sign in (x - alpha)^2: if the standard form is (x + 3)^2, then alpha = -3 and not 3, because (x + 3)^2 = (x - (-3))^2.