Domain and Range of Linear & Quadratic Functions

Once you can read domain and range off any graph, lines and parabolas become the easy cases — their answers follow a pattern you can memorize and then verify with a glance. A full line with arrows on both ends covers everything, and a parabola is unlimited side to side but capped in one vertical direction by its vertex.

The skill here is knowing the pattern and knowing why it works, so a test question can't shake you by changing the numbers. Two facts carry the whole lesson: arrows mean the graph goes forever, and a parabola's vertex is either its lowest or highest point.

Linear functions: everything, both ways

A linear function like $f(x) = 2x - 1$ graphs as a line with arrows on both ends. Left to right it never stops, so the domain is all real numbers. Because the line is slanted, it also climbs forever and drops forever, so the range is all real numbers too. In interval notation, both are $(-\infty, \infty)$ .

The one exception worth knowing: a horizontal line like $f(x) = 3$ still has domain all real numbers, but it only ever reaches one height, so its range is just $\{3\}$ .

Quadratic functions: the vertex sets the range

A quadratic function graphs as a parabola. Side to side, a parabola never stops — you can square any number — so the domain of every quadratic is all real numbers.

The range is where the vertex takes over. If the parabola opens upward, the vertex is the lowest point, and the range is every $y$ from the vertex's height on up: $y \geq k$ . If it opens downward, the vertex is the highest point, and the range is $y \leq k$ . The parabola below has vertex $(2, 1)$ and opens upward, so its range is $y \geq 1$ .

The two-question routine

For any line or parabola, ask: does the graph ever stop going left or right? For both families with arrows, no — domain is all real numbers. Then ask: does the graph have a lowest or highest point? A slanted line has neither, so its range is all real numbers. A parabola has exactly one, the vertex, so its range starts or ends there. Only the $y$ -coordinate of the vertex matters for range — the $x$ -coordinate never appears in the answer.

Worked examples

Example 1: a slanted line

Find the domain and range of $f(x) = -3x + 4$ .

The graph is a line with arrows on both ends

Left to right it never stops

\text{domain: all real numbers}

It climbs and drops without limit

\text{range: all real numbers}

Answer: Domain $(-\infty, \infty)$ , range $(-\infty, \infty)$

Example 2: a parabola opening upward

A parabola opens upward with vertex $(3, -2)$ . Find the domain and range.

Every quadratic accepts every input

\text{domain: all real numbers}

Opening upward means the vertex is the minimum point

The lowest output is the vertex's

y

-value,

-2

y \geq -2

Answer: Domain $(-\infty, \infty)$ , range $y \geq -2$ , or $[-2, \infty)$

Example 3: a parabola opening downward

A parabola opens downward with vertex $(-1, 6)$ . Find the domain and range.

Domain is still all real numbers

(-\infty, \infty)

Opening downward means the vertex is the maximum point

The highest output is

6

, and every

y

below it is reached

y \leq 6

Answer: Domain $(-\infty, \infty)$ , range $y \leq 6$ , or $(-\infty, 6]$

Try one yourself

Common questions

Why is the domain of a parabola all real numbers?

Because you can square any number. A quadratic rule like $f(x) = x^{2} - 3$ produces an output no matter what input you hand it, so no $x$ -value is ever excluded.

How do I tell if the vertex is a minimum or a maximum?

Look at which way the parabola opens. Opening upward makes the vertex the lowest point (a minimum), so the range is $y \geq k$ . Opening downward makes it the highest point (a maximum), so the range is $y \leq k$ .

Does the x-coordinate of the vertex affect the range?

No. The range only measures heights, so only the vertex's $y$ -coordinate shows up. A vertex at $(7, -2)$ and a vertex at $(-100, -2)$ give the same range for an upward parabola: $y \geq -2$ .

Do lines ever have a limited range?

Only horizontal ones. A horizontal line like $f(x) = 5$ stays at one height forever, so its range is just $5$ . Any slanted line reaches every height, giving a range of all real numbers.

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Domain and Range of Linear & Quadratic Functions

Linear functions: everything, both ways

Quadratic functions: the vertex sets the range

The two-question routine

Worked examples

Example 1: a slanted line

Example 2: a parabola opening upward

Example 3: a parabola opening downward

Try one yourself

Common questions

Keep learning

Domain and Range of Continuous Functions

Intro to Domain and Range

Shapes of Graphs

Graphing Functions