What Is Domain and Range of a Circle Graph?
A circle with center (h, k) and radius r reaches x values from h − r to h + r, and y values from k − r to k + r. Those two intervals are its domain and range.
If your equation is in general form, x² + y² + Dx + Ey + F = 0, the calculator completes the square first to find the center and radius. Then it reads off the domain and range.
A circle is not a function of x, because most x values give two points on the circle. Only a semicircle passes the vertical line test.
How to Use the Domain and Range of a Circle Graph Calculator
- Choose center and radius, or general form.
- Enter the values.
- Read the domain and range in interval notation and check the graph.
Domain and Range of a Circle Graph Formula
Circle
Domain and range
Domain and Range of a Circle Graph Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| (h, k) | Center of the circle | |
| r | Radius | |
| Domain | [h − r, h + r] | All x values on the circle |
| Range | [k − r, k + r] | All y values on the circle |
Domain and Range of a Circle Graph Worked Example
With the example values loaded in the calculator above, the steps are:
- Equation: (x − 2)² + (y − -1)² = 3².
- Domain = [h − r, h + r] = [-1, 5].
- Range = [k − r, k + r] = [-4, 2].
Result: Domain = [-1, 5]; Range = [-4, 2].
Worked example: center (2, −1), radius 3
Domain = [2 − 3, 2 + 3] = [−1, 5]. Range = [−1 − 3, −1 + 3] = [−4, 2].
The leftmost point of the circle is at x = −1 and the rightmost at x = 5, and the lowest and highest points are at y = −4 and y = 2.
From the general form
If the equation is x² + y² + Dx + Ey + F = 0, complete the square. The center is (−D/2, −E/2) and r² = (D/2)² + (E/2)² − F. For D = −4, E = 2 and F = −4 this gives center (2, −1) and radius 3.
Where Domain and Range of a Circle Graph Is Used
Algebra homework
Write the domain and range of a circle in interval notation.
Graphing circles
Find the extreme left, right, top and bottom points.
Completing the square
Convert the general equation to center and radius form.
Common Domain and Range of a Circle Graph Mistakes to Avoid
- Use square brackets because the end points are on the circle.
- If r² comes out zero or negative there is no circle to graph.
Domain and Range of a Circle Graph Terms and Definitions
- Domain
- All x values the graph reaches.
- Range
- All y values the graph reaches.
Domain and Range of a Circle Graph FAQs
What is the domain of a circle?
The interval from h − r to h + r, where h is the x coordinate of the center and r the radius.
What is the range of a circle?
The interval from k − r to k + r.
Is a circle a function?
No. A vertical line can cross it twice. Only a semicircle is a function.
How do I find the center from a general equation?
Complete the square. The center is (−D/2, −E/2).