Examine the sign and plot the parabola

Investigating the sign of f

The standard form f(x) = a(x - alpha)² + beta allows us to examine the sign of f without further factorisation, by reducing it to the inequality a(x - alpha)² ≥ -beta, following the same three cases as for solving the equation.

Method

1. Calculate any roots x₁ and x₂ as in the previous lesson.
2. If there are two roots, f(x) has the same sign as a outside the interval [x₁; x₂] and the opposite sign between the roots.
3. If there are no roots, f retains the sign of a over the entire set R.

Example

For f(x) = 2(x - 2)² - 3, the roots are x₁ = 2 - √(3/2) and x₂ = 2 + √(3/2). As a = 2 > 0: f(x) < 0 between x1 and x2, f(x) > 0 outside this interval, and f(x1) = f(x2) = 0.

Plotting the shape of the parabola

To sketch the curve quickly: plot the vertex S(alpha; beta), draw the axis of symmetry x = alpha, orient the branches upwards if a > 0 or downwards if a < 0, then plot the roots if they exist.

Common pitfall

Do not confuse the sign of f with the sign of a: between the roots, f takes the opposite sign to that of a, not the same one.