Domain and Range of Quadratic Functions

You have a quadratic problem and need the domain and range. The domain of every quadratic function is all real numbers, (−∞, ∞). The range depends entirely on the vertex and whether the curve opens up or down. Find the range using the vertex method, from standard form, or from a graph.

For any quadratic function f(x) = ax² + bx + c, the domain is always all real numbers, written as (−∞, ∞) in interval notation. There is no denominator to cause a zero and no even-index radical to restrict the radicand. This never changes, regardless of the values of a, b, or c. It is a polynomial function of degree two, and all polynomial functions have this domain. As OpenStax Precalculus 2e section 3.2 explains, the domain is the set of all possible input values, and here every real x works.

Vertex Form Is Fastest

In f(x) = a(x−h)² + k, the vertex is (h, k). The sign of a tells you the direction. If a > 0, the curve opens upward and the range is [k, ∞). If a < 0, it opens downward and the range is (−∞, k]. The y-coordinate of the vertex becomes the minimum or maximum, and the other side is unbounded.

For example, f(x) = −2(x + 1)² + 3 has vertex (−1, 3). Because a = −2 is negative, the curve opens down. The range is (−∞, 3]. The domain is still (−∞, ∞).

If the quadratic is in standard form ax² + bx + c, find the vertex x‑coordinate using h = −b/(2a). Then compute k = f(h). This gives you the same (h, k) as vertex form. The range rule is identical: if a > 0, range is [k, ∞); if a < 0, range is (−∞, k].

Take f(x) = x² − 4x + 3. Here a = 1, b = −4. So h = −(−4)/(2·1) = 2. Then k = f(2) = 2² − 4·2 + 3 = 4 − 8 + 3 = −1. Since a > 0, the range is [−1, ∞). OpenStax Precalculus 2e section 3.2 shows this exact example. The domain remains all reals.

When you have a graph but no equation, read the range directly off the y‑axis. Find the lowest or highest y‑value that the curve reaches. That is the vertex y‑coordinate. If the curve opens up, the range goes from that y‑value upward to infinity. If it opens down, it goes from negative infinity up to that y‑value. Use interval notation: a bracket includes the vertex value, a parenthesis means infinity is never reached.

Example 1: Opens Upward

f(x) = 3x² − 6x + 1. h = −(−6)/(2·3) = 6/6 = 1. k = f(1) = 3(1)² − 6(1) + 1 = 3 − 6 + 1 = −2. a > 0, so range is [−2, ∞). Domain is (−∞, ∞).

Example 2: Opens Downward

f(x) = −x² + 4x − 7. h = −4/(2·(−1)) = −4/(−2) = 2. k = −(2)² + 4(2) − 7 = −4 + 8 − 7 = −3. a < 0, so range is (−∞, −3]. Domain is (−∞, ∞).

Example 3: Vertex Form

f(x) = 5(x − 3)² − 8. Vertex is (3, −8). a > 0, so range is [−8, ∞). Domain is (−∞, ∞).

Sometimes a problem gives you only part of the curve, like the interval x ∈ [0, 5]. Then the domain of that restricted quadratic is that interval, not all reals. The range is no longer just the vertex y‑value. Evaluate the function at both endpoints and at the vertex if it falls inside the interval. The smallest y‑value among those points becomes the lower bound of the range, the largest becomes the upper bound. This applies to problems where the domain and range are limited by context, such as projectile height over a specific time window.

Work these on your own. 1) f(x) = 2x² + 12x + 10. Find the range using the vertex method. 2) f(x) = −3(x − 4)² + 1. Write the domain and range. 3) f(x) = x² − 2x + 5 with domain [−1, 3]. Find the range. 4) f(x) = −0.5x² + 2x − 3. Determine the vertex and the range. Check your work against the rules above. If you struggle on number 3, remember that a restricted domain means the range might not start at the vertex if the vertex is outside the given x‑interval.

Direction	Sign of a	Vertex Role	Range (Interval Notation)
Opens upward	a > 0	Vertex y‑coordinate is minimum	[k, ∞)
Opens downward	a < 0	Vertex y‑coordinate is maximum	(−∞, k]

This topic suits Algebra 1 students learning to distinguish allowed x from possible y for the first time, Algebra 2 students handling quadratic functions, and precalculus students who need interval notation. Teachers and tutors who want a reliable method to check student work will also find it useful. Skip this if you need calculus‑level continuity analysis or epsilon‑delta proofs; go to a calculus textbook or Paul's Online Math Notes instead. Also skip if you cannot evaluate a function at a given x‑value; first master function notation from OpenStax College Algebra 2e, Section 3.1.