1-9 Sign of a function

The concept

• f is positive on a set of values of x when f(x) > 0, and negative when f(x) < 0; the zeros are the values that make f(x) = 0.

Sign of a linear function ax + b

• If a > 0 the function is negative to the left of its zero −b/a and positive to the right of it.
• If a < 0 the signs are reversed: positive on the left, negative on the right.

Sign of a quadratic ax² + bx + c with two different real zeros α < β

• If a > 0: positive outside the two zeros (x < α or x > β) and negative between them.
• If a < 0: negative outside and positive between.

When Δ ≤ 0

• Δ = 0: ax² + bx + c = a(x − r)² with r = −b/(2a), so the sign is the sign of a for all x except the zero at r.
• Δ < 0: the expression never changes sign; it keeps the sign of a for every real x.

Sign from the graph

• The x-intercepts of the curve are the zeros of the function; above the x-axis the function is positive, below it is negative.