Monotonicity and range
Learning Objectives
- State and prove the monotonicity of a quadratic function on each side of the vertex.
- Read the monotonicity intervals from the sign of \(a\) and the abscissa \(x_V\).
- Build the variation table of a quadratic function.
- Determine the range \(\operatorname{Im} f\): \([y_V, \infty)\) when \(a > 0\), \((-\infty, y_V]\) when \(a < 0\).
- Explain why a quadratic function is not monotonic — nor injective — on all of \(\mathbb{R}\), but is on each branch.
The whole story of the values
We now know the shape of the parabola and its special point, the vertex. Two questions about the values of \(f(x) = ax^2 + bx + c\) remain, and together they complete its portrait:
As \(x\) moves along the real line, where does \(f\) rise and where does it fall? — its monotonicity.
Taken all together, which values does \(f\) actually reach? — its range, or image \(\operatorname{Im} f\).
Both concepts come from the general theory of functions; here we pin them down exactly for the quadratic, and the answer to each is written on the vertex. The canonical form \(f(x) = a(x - x_V)^2 + y_V\) — where every step of the argument will live — does the work once again.
Monotonicity of the quadratic function
Let \(f(x) = ax^2 + bx + c\) with vertex abscissa \(x_V = -\dfrac{b}{2a}\).
If \(a > 0\), then \(f\) is strictly decreasing on \((-\infty, x_V]\) and strictly increasing on \([x_V, \infty)\).
If \(a < 0\), then \(f\) is strictly increasing on \((-\infty, x_V]\) and strictly decreasing on \([x_V, \infty)\).
The vertex is the single point where the function turns.
Take any two inputs \(x_1 < x_2\) and compare their values through the canonical form. Since the constant \(y_V\) cancels,
\[f(x_2) - f(x_1) = a\bigl[(x_2 - x_V)^2 - (x_1 - x_V)^2\bigr].\]
The bracket is a difference of squares, \(u^2 - v^2 = (u - v)(u + v)\), with \(u = x_2 - x_V\) and \(v = x_1 - x_V\):
\[f(x_2) - f(x_1) = a\,(x_2 - x_1)\,(x_1 + x_2 - 2x_V).\]
The middle factor \(x_2 - x_1\) is positive, since \(x_1 < x_2\). So the sign of \(f(x_2) - f(x_1)\) is the sign of \(a\,(x_1 + x_2 - 2x_V)\).
Suppose \(a > 0\). If both inputs lie in \((-\infty, x_V]\), then \(x_1 + x_2 < 2x_V\), so the last factor is negative and \(f(x_2) < f(x_1)\): the function decreases. If both lie in \([x_V, \infty)\), then \(x_1 + x_2 > 2x_V\), the last factor is positive, and \(f(x_2) > f(x_1)\): it increases. When \(a < 0\) every sign flips, exchanging the two behaviours.
The variation table
The monotonicity is captured at a glance by a variation table: the top row lists the inputs from \(-\infty\) to \(+\infty\), marking the turning point \(x_V\); the bottom row draws the movement of \(f(x)\) with arrows that meet at the extreme value \(y_V\). The sign of \(a\) flips the whole table:
Each table is the parabola's two branches compressed to a single line: the arrow descends to \(y_V\) and climbs again when \(a > 0\), and climbs to \(y_V\) then descends when \(a < 0\).
Reading the variation
For \(f(x) = x^2 - 6x + 5\): here \(a = 1 > 0\) and \(x_V = -\dfrac{-6}{2} = 3\), \(y_V = f(3) = 9 - 18 + 5 = -4\). So \(f\) is strictly decreasing on \((-\infty, 3]\) and strictly increasing on \([3, \infty)\), turning at \(V(3, -4)\).
The values bear it out: \(f(0) = 5,\; f(1) = 0,\; f(2) = -3,\; f(3) = -4,\; f(4) = -3,\; f(5) = 0\) — falling steadily to \(-4\), then climbing. Notice \(f(2) = f(4)\) and \(f(1) = f(5)\): the two branches take the same values in reverse, the mirror symmetry seen once more.
The range of the quadratic function
Let \(f \colon \mathbb{R} \to \mathbb{R},\; f(x) = ax^2 + bx + c\), with vertex ordinate \(y_V = -\dfrac{\Delta}{4a}\).
If \(a > 0\), then \(\operatorname{Im} f = [y_V, \infty)\): the function attains every value \(\geq y_V\) and no value below \(y_V\).
If \(a < 0\), then \(\operatorname{Im} f = (-\infty, y_V]\): every value \(\leq y_V\), and none above.
Take \(a > 0\); the case \(a < 0\) is identical with the inequalities reversed. Two inclusions are needed.
No value below \(y_V\) is reached. By the extremum theorem of Lesson 1, \(f(x) = a(x - x_V)^2 + y_V \geq y_V\) for every \(x\), so every value of \(f\) lies in \([y_V, \infty)\).
Every value \(\geq y_V\) is reached. Fix any real \(y \geq y_V\); we exhibit an \(x\) with \(f(x) = y\). Solving \(a(x - x_V)^2 + y_V = y\) gives
\[(x - x_V)^2 = \frac{y - y_V}{a}.\]
The right-hand side is \(\geq 0\), because \(y - y_V \geq 0\) and \(a > 0\), so it has a square root: taking \(x = x_V + \sqrt{\dfrac{y - y_V}{a}}\) yields \(f(x) = y\). Thus every \(y \geq y_V\) is a value of \(f\), and \(\operatorname{Im} f = [y_V, \infty)\).
Range, both directions
Upward. For \(f(x) = x^2 - 6x + 5\) above, \(y_V = -4\) and \(a > 0\), so \(\operatorname{Im} f = [-4, \infty)\): the function reaches \(-4\) (at \(x = 3\)) and everything above it, but never dips below \(-4\).
Downward. For \(g(x) = -x^2 + 4x + 1\): \(a = -1 < 0\), \(x_V = 2\), \(y_V = g(2) = -4 + 8 + 1 = 5\), so \(\operatorname{Im} g = (-\infty, 5]\): every value up to and including \(5\), none above.
The range is the shadow the parabola casts on the \(y\)-axis: a single ray, starting at \(y_V\), pointing the way the branches open.
Connection to Computer Science
When a program must summarize many measurements \(x_1, \dots, x_n\) by a single number \(m\), a common choice is the one that makes the total squared error \(g(m) = (x_1 - m)^2 + \dots + (x_n - m)^2\) as small as possible. Multiplied out, \(g\) is a quadratic function of \(m\) with positive leading coefficient, so its minimum sits at the vertex \(m = -\tfrac{b}{2a}\) — which works out to be exactly the average of the measurements. The everyday rule "take the mean" is, underneath, "jump to the bottom of a parabola", and this is why averaging is the error-minimizing summary used throughout statistics and machine learning.
Monotonic, injective — on what?
A quick question exposes a common confusion: is the quadratic function increasing, or decreasing? The honest answer is neither — not on all of \(\mathbb{R}\). It decreases on one side of the vertex and increases on the other, so no single label fits the whole line. This is a genuine break from the linear function, which was monotonic everywhere. Monotonicity for a quadratic is always relative to an interval, and the interval must be named.
The same split has a sharper consequence, recalling the general theory. Because \(f(x_V - t) = f(x_V + t)\), the function takes each value above its minimum twice — so \(f\) is not injective on \(\mathbb{R}\), and therefore has no inverse there. For example \(x^2 = 9\) has the two solutions \(3\) and \(-3\): the rule "square it" cannot be undone without knowing which sign we started from.
Yet restrict \(f\) to a single branch — say \([x_V, \infty)\), where it is strictly increasing — and it becomes injective, hence invertible, on that branch. This is exactly how the square-root function arises: it is the inverse of \(x \mapsto x^2\) restricted to \([0, \infty)\), which is why \(\sqrt{\phantom{x}}\) returns only the nonnegative root. A function too symmetric to invert whole becomes invertible the moment we look at one branch of it.
Exercises
Exercise 1
Consider \(f(x) = x^2 - 8x + 10\).
Exercise 2
Determine \(\operatorname{Im} f\) for each quadratic function.
Exercise 3
Decide whether each statement about monotonicity and range is true or false.
Exercise 4
Consider \(f(x) = -x^2 + 6x - 5\) (note \(a = -1 < 0\)).
Exercise 5
From the sign of \(a\), decide how each function moves as \(x\) goes from \(-\infty\) to \(+\infty\).
Summary
Monotonicity: a quadratic function turns exactly at the vertex. For \(a > 0\) it is decreasing on \((-\infty, x_V]\) and increasing on \([x_V, \infty)\); for \(a < 0\), the reverse. The variation table records this in one line.
Range: \(\operatorname{Im} f = [y_V, \infty)\) when \(a > 0\), and \((-\infty, y_V]\) when \(a < 0\) — a half-line starting at the extreme value \(y_V\).
Both facts are proved from the canonical form: monotonicity from the sign of \(a(x_1 + x_2 - 2x_V)\), the range from \(a(x - x_V)^2 \geq 0\) plus solving for any target value.
A quadratic is not monotonic and not injective on \(\mathbb{R}\) — but on a single branch it is both, which is how the square-root function is born as the inverse of a branch of \(x^2\).
With this, Unit 2 is complete: definition, canonical form, graph, and full variation of the quadratic function. Unit 3 turns to its sign, and the inequalities and intersections that follow.