StatisticsCalc

Domain and Range of Parabola Calculator

This domain and range of parabola calculator gives the vertex, axis, intercepts, domain and range in interval notation for any quadratic.

Checks on your data

  • The vertex was found with −b ÷ 2a and then substituted back into the equation.
  • A parabola never has a restricted domain on its own unless a problem states one. Word problems often do, so check the context.

Results

Domain
(−∞, ∞)
Range
[-1, ∞)
Vertex
(2, -1)
Axis of symmetry
x = 2
Opens
up
Discriminant (b² − 4ac)
4
x-intercepts
1, 3
y-intercept
3
Parabola with its vertex marked
-1.05-1-0.951.91.9522.052.1xy
Show step-by-step solution
  1. Equation: y = 1x² − 4x + 3.
  2. Vertex coordinate = −b ÷ (2a) = 2. The other vertex coordinate is -1.
  3. y at the vertex = -1. Since a > 0, the parabola opens up.
  4. Every real x is allowed, so the domain is (−∞, ∞). y stops at the vertex, so the range is [-1, ∞).
Vertex xh = −b / 2a
Opens uprange [k, ∞)
Opens downrange (−∞, k]

What Is Domain and Range of Parabola?

The domain of a parabola y = ax² + bx + c is every real number. Its range is bounded at the vertex: it starts at the vertex y value if the parabola opens up, and ends there if it opens down.

To find the vertex, use x = −b ÷ 2a and substitute back to get y. The sign of a tells you the direction. For a parabola that opens sideways, x = ay² + by + c, the roles of domain and range swap.

Choose the form, enter a, b and c, and read the answer in interval notation.

How to Use the Domain and Range of Parabola Calculator

  1. Choose the form of the equation.
  2. Enter a, b and c.
  3. Read the vertex, domain and range, then check the graph.

Domain and Range of Parabola Formula

Vertex

h=−b2a,k=c−b24ah=-\frac{b}{2a},\quad k=c-\frac{b^2}{4a}

Range (opens up)

[k,∞)[k,\infty)

Domain and Range of Parabola Variables Explained

SymbolMeaningNotes
aCoefficient of the squared termSign sets the direction
(h, k)Vertex of the parabolah = −b / 2a
DomainAll allowed input values
RangeAll output values reached[k, ∞) if a > 0

Domain and Range of Parabola Worked Example

With the example values loaded in the calculator above, the steps are:

  1. Equation: y = 1x² − 4x + 3.
  2. Vertex coordinate = −b ÷ (2a) = 2. The other vertex coordinate is -1.
  3. y at the vertex = -1. Since a > 0, the parabola opens up.
  4. Every real x is allowed, so the domain is (−∞, ∞). y stops at the vertex, so the range is [-1, ∞).

Result: Domain = (−∞, ∞); Range = [-1, ∞).

Worked example: y = x² − 4x + 3

The vertex x is −(−4) ÷ (2 × 1) = 2. Then y = 4 − 8 + 3 = −1, so the vertex is (2, −1).

Since a = 1 is positive, the parabola opens up. The domain is (−∞, ∞) and the range is [−1, ∞).

Parabolas that open sideways

For x = ay² + by + c the roles swap. The range is all real numbers and the domain starts or ends at the vertex x value.

Where Domain and Range of Parabola Is Used

Algebra homework

Write the domain and range in interval notation.

Graphing quadratics

Locate the vertex, axis and intercepts before sketching.

Word problems

Find the maximum or minimum value from the vertex.

Common Domain and Range of Parabola Mistakes to Avoid

  • Closed bracket at the vertex value, round bracket at infinity.
  • The sign of a tells you the direction.

Domain and Range of Parabola Terms and Definitions

Vertex
The turning point of the parabola, its minimum or maximum.
Interval notation
A way to write a set of numbers, such as [−1, ∞).

Domain and Range of Parabola FAQs

What is the domain of a parabola?

For y = ax² + bx + c it is all real numbers, (−∞, ∞).

How do you find the range of a parabola?

Find the y value of the vertex. If a > 0 the range is [k, ∞). If a < 0 it is (−∞, k].

What about a parabola that opens sideways?

Then the range is all reals and the domain is bounded by the vertex x value.

Why is a not allowed to be 0?

With a = 0 the equation is linear, so it is not a parabola.

Last updated 2026-11-09. See our disclaimer.