The sign of the quadratic function
Learning Objectives
- State the sign theorem for the quadratic function, by cases on \(\Delta\).
- Prove the key fact: outside the roots \(f\) has the sign of \(a\); between them, the opposite sign.
- Build the sign table of a quadratic function.
- Recognize when a quadratic keeps one constant sign for all \(x\) (\(\Delta < 0\), or \(\Delta = 0\) except at the vertex).
- Read the sign of \(f\) off its parabola, matching algebra to picture.
A new question about an old function
Unit 1 asked when is \(ax^2 + bx + c\) equal to zero? — and answered it with the discriminant. Unit 2 drew the function and studied where it rises, falls, and how high or low it goes. One natural question remains, and it is the key that unlocks the rest of this unit:
For which \(x\) is \(f(x)\) positive, and for which is it negative?
This is the sign of the function. Geometrically it is transparent: \(f(x) > 0\) exactly where the parabola runs above the \(Ox\) axis, \(f(x) < 0\) where it runs below, and \(f(x) = 0\) at the crossing points — the roots. So the whole answer is governed by how the parabola sits relative to \(Ox\), which Unit 1's discriminant already decides. We now turn that picture into an exact rule.
Three positions of the parabola
Recall from Lesson 2 of Unit 2 how a parabola can meet the \(Ox\) axis, according to \(\Delta\): in two points, in one (a touch at the vertex), or in none. Take \(a > 0\) for the moment, so the branches open upward:
The pictures already suggest the whole theorem. With two roots the upward parabola dips below the axis between them and stays above it outside; with a double root it only touches, never going negative; with no root it floats entirely above. Reversing to \(a < 0\) flips every parabola upside down, exchanging "above" and "below". It remains to prove what the eye proposes.
The sign of the quadratic function
Let \(f(x) = ax^2 + bx + c\) with \(a \neq 0\) and \(\Delta = b^2 - 4ac\).
\(\Delta > 0\) (roots \(x_1 < x_2\)): \(f(x)\) has the sign of \(a\) for \(x < x_1\) and for \(x > x_2\), the sign opposite to \(a\) for \(x_1 < x < x_2\), and \(f(x) = 0\) at \(x_1, x_2\).
\(\Delta = 0\) (double root \(x_0\)): \(f(x)\) has the sign of \(a\) for every \(x \neq x_0\), and \(f(x_0) = 0\).
\(\Delta < 0\) (no real roots): \(f(x)\) has the sign of \(a\) for all \(x \in \mathbb{R}\) — it never vanishes and never changes sign.
In one phrase: a quadratic takes the sign of \(a\) everywhere, except strictly between distinct roots, where it takes the opposite sign.
Everything follows from the canonical form of Unit 2, \(f(x) = a\left[(x - x_V)^2 + \dfrac{-\Delta}{4a^2}\right]\), but it is cleanest to split by cases.
Case \(\Delta < 0\). Write the canonical form as \(f(x) = a\Bigl[(x - x_V)^2 + \dfrac{-\Delta}{4a^2}\Bigr]\). Since \(\Delta < 0\), the term \(\dfrac{-\Delta}{4a^2}\) is strictly positive, so the bracket is a sum of a square and a positive number — strictly positive for every \(x\). Thus \(f(x)\) is \(a\) times a positive quantity: it has the sign of \(a\) throughout, and never vanishes.
Case \(\Delta = 0\). Here \(f(x) = a(x - x_0)^2\) with \(x_0 = x_V\) (the factorization from Unit 1). The square \((x - x_0)^2\) is positive for \(x \neq x_0\) and zero at \(x_0\), so \(f\) has the sign of \(a\) everywhere except at \(x_0\), where it is \(0\).
Case \(\Delta > 0\). Now \(f(x) = a(x - x_1)(x - x_2)\) with \(x_1 < x_2\). The sign of \(f(x)\) is the sign of \(a\) times the sign of the product \((x - x_1)(x - x_2)\). Examine that product:
for \(x < x_1\): both factors are negative, product positive;
for \(x_1 < x < x_2\): the first factor is positive, the second negative, product negative;
for \(x > x_2\): both factors positive, product positive.
So \((x - x_1)(x - x_2)\) is positive outside \([x_1, x_2]\) and negative strictly inside; multiplying by \(a\) gives the stated signs.
The sign table
The result is recorded compactly in a sign table: the top row marks the roots along the real line, and the bottom row gives the sign of \(f(x)\) in each resulting interval, with a \(0\) at each root. For \(f(x) = x^2 - x - 6\) — where \(a = 1 > 0\) and, since \(x^2 - x - 6 = (x + 2)(x - 3)\), the roots are \(-2\) and \(3\):
Reading it: \(f(x) > 0\) for \(x \in (-\infty, -2) \cup (3, \infty)\), \(f(x) < 0\) for \(x \in (-2, 3)\), and \(f(x) = 0\) at \(x \in \{-2, 3\}\). The sign of \(a\) (here \(+\)) appears in the outer intervals; the opposite sign sits between the roots. This little table is the engine of the next lesson's inequalities.
All three cases, once each
Two roots, \(a < 0\). For \(f(x) = -x^2 + 4x - 3 = -(x - 1)(x - 3)\): roots \(1\) and \(3\), and \(a = -1 < 0\). Outside \([1, 3]\), \(f\) has the sign of \(a\): negative. Between \(1\) and \(3\): positive. So \(f(x) > 0\) on \((1, 3)\) and \(f(x) < 0\) on \((-\infty, 1) \cup (3, \infty)\).
Double root. For \(f(x) = x^2 - 6x + 9 = (x - 3)^2\): \(\Delta = 0\), and \(f(x) \geq 0\) always, with equality only at \(x = 3\). It is positive everywhere except the single point where it touches the axis.
No root. For \(f(x) = x^2 + x + 1\): \(\Delta = 1 - 4 = -3 < 0\) and \(a > 0\), so \(f(x) > 0\) for every real \(x\) — a quadratic that is strictly positive everywhere. (This is the same expression whose non-existent roots we discussed in Unit 1; now we see it never even reaches zero.)
Connection to Computer Science
When two moving objects in a physics engine — game characters, simulated particles, spacecraft in orbit software — might collide, the squared distance between them over time works out to be a quadratic function \(f(t)\). The objects overlap exactly during the interval where \(f(t)\) drops below a threshold, i.e. where a certain quadratic is negative: the collision's start and end times are the two roots, and the "are they touching right now?" test is a sign check. Detecting the collision is reading the sign of a quadratic — the very table of this lesson, evaluated thousands of times a second.
What "positive between the roots" quietly assumes
The slogan "the sign of \(a\) outside the roots, the opposite sign between them" is easy to chant — and easy to misapply, because the word "between" presumes there are two roots to be between. Two traps follow.
First, when \(\Delta \leq 0\) there is no interval "between the roots" at all: the sign never flips. A student who mechanically writes "\(+\;0\;-\;0\;+\)" for every quadratic will invent a sign change that does not exist. The sign table must be built after checking \(\Delta\) — existence of the roots comes first, exactly the discipline of Unit 1.
Second, the rule speaks of the sign of \(a\), not the sign of the constant term or of \(x^2\)'s coefficient read carelessly. It is a common slip to fix the sign from \(f(0) = c\); but \(c\) is only the value at a single point, and it can differ from the "outer" sign \(a\). For \(f(x) = x^2 - x - 6\) we have \(a = 1 > 0\) yet \(c = -6 < 0\), because \(0\) lies between the roots. The controlling quantity outside the roots is always \(a\).
The habit of rigor: name \(a\) and \(\Delta\) before writing a single sign, and let the number of roots decide how many sign changes the table is even allowed to have.
Exercises
Exercise 1
Let \(f(x) = x^2 - x - 6 = (x + 2)(x - 3)\), with roots \(-2\) and \(3\) and \(a = 1 > 0\). Sort each input by the sign of \(f\) there.
Exercise 2
For each function, choose the set where \(f(x) < 0\).
Exercise 3
Decide whether each statement about the sign of a quadratic function is true or false.
Exercise 4
For \(f(x) = 2x^2 - 2x - 12\), find the two values of \(x\) where the sign of \(f\) changes (its roots), smaller first.
Exercise 5
Decide whether each function keeps one sign for all \(x\) (never crossing \(Ox\)), or changes sign.
Summary
The sign of \(f(x) = ax^2 + bx + c\) tracks where its parabola runs above \(Ox\) (\(f > 0\)), below (\(f < 0\)), or crosses it (\(f = 0\)).
The rule: \(f\) takes the sign of \(a\) everywhere, except strictly between two distinct roots, where it takes the opposite sign — proved from the factored and canonical forms.
By cases on \(\Delta\): \(\Delta > 0\) — sign of \(a\), opposite, sign of \(a\); \(\Delta = 0\) — sign of \(a\) everywhere but the double root; \(\Delta < 0\) — the sign of \(a\) on all of \(\mathbb{R}\).
The sign table records this along the real line: roots on top, signs (and zeros) below. Check \(\Delta\) first — with no distinct roots there is no sign change to record.
Next: turning the sign table into a method for solving quadratic inequalities.