Graphing Circles & Writing Equations
Master standard form, graphing from an equation, writing equations from a graph, and converting general form using completing the square.
🎯 By the end of this lesson you will be able to:
- Identify the center and radius from standard form \((x-h)^2 + (y-k)^2 = r^2\)
- Graph a circle by plotting the center and four compass points
- Write the equation of a circle given a graph
- Convert general form to standard form by completing the square
📋 In This Lesson
1. Standard Form of a Circle
💡 What is Standard Form?
A circle is the set of all points that are exactly \(r\) units away from a fixed center point \((h, k)\). That definition leads directly to the standard form equation:
$$(x – h)^2 + (y – k)^2 = r^2$$
- \((h, k)\) — the center of the circle
- \(r\) — the radius (always positive)
Notice the signs: the formula subtracts h and k. If the center is \((-3, 5)\), the equation reads \((x+3)^2 + (y-5)^2 = r^2\).
🟢 Reading Center & Radius at a Glance
Given \((x – 2)^2 + (y + 1)^2 = 25\):
- Center: \((2,\ -1)\) — flip the signs inside each parenthesis
- Radius: \(r = \sqrt{25} = 5\)
✏️ Practice Problems
1. Identify the center and radius of \((x-4)^2 + (y-3)^2 = 16\).
👁️ Show Solution
Center: \((4, 3)\) | Radius: \(r = \sqrt{16} = 4\)
2. Identify the center and radius of \((x+5)^2 + (y-2)^2 = 9\).
👁️ Show Solution
Center: \((-5, 2)\) | Radius: \(r = \sqrt{9} = 3\)
3. Write the equation of a circle with center \((0, 0)\) and radius \(7\).
👁️ Show Solution
$$x^2 + y^2 = 49$$
2. How to Graph a Circle
💡 The 5-Step Method
- Write the equation in standard form and identify \((h, k)\) and \(r\)
- Plot the center \((h, k)\)
- From the center, count \(r\) units right, left, up, and down — mark those 4 compass points
- Sketch a smooth curve through all four points
- Label the center and equation
🎬 Watch: How to Graph a Circle — center, radius, and compass points step by step
✏️ Worked Example
Graph \((x-2)^2 + (y-1)^2 = 9\)
👁️ Show Solution
Step 1: Already in standard form. Center \((2, 1)\), radius \(r = 3\).
Step 2: Plot center at \((2, 1)\).
Step 3: Mark compass points:
- Right: \((5, 1)\)
- Left: \((-1, 1)\)
- Up: \((2, 4)\)
- Down: \((2, -2)\)
Step 4: Connect with a smooth curve.
Practice. Graph \((x+1)^2 + (y-3)^2 = 4\).
👁️ Show Solution
Center: \((-1, 3)\), radius \(r = 2\).
Compass points: \((1,3)\), \((-3,3)\), \((-1,5)\), \((-1,1)\)
3. Writing the Equation from a Graph
💡 Reading a Circle from a Graph
- Identify the center \((h, k)\) from the graph
- Count the radius \(r\) from the center to any point on the circle
- Substitute into \((x-h)^2 + (y-k)^2 = r^2\)
- Watch the signs — subtracting a negative becomes addition
🎬 Watch: Writing the Equation of a Circle from a Graph
✏️ Worked Example
Write the equation for a circle with center \((2, -1)\) and radius \(3\).
👁️ Show Solution
Substitute \(h=2,\ k=-1,\ r=3\):
$$(x-2)^2 + (y-(-1))^2 = 3^2$$
$$(x-2)^2 + (y+1)^2 = 9$$
Note: \(y – (-1)\) simplifies to \(y + 1\)
Practice 1. Center \((-3, 4)\), radius \(5\).
👁️ Show Solution
$$(x+3)^2 + (y-4)^2 = 25$$
Practice 2. Center \((0, -2)\), radius \(6\).
👁️ Show Solution
$$x^2 + (y+2)^2 = 36$$
4. Completing the Square
💡 Why Do We Need This?
Sometimes a circle equation is given in general form:
$$x^2 + y^2 + Dx + Ey + F = 0$$
To graph it, we need to convert it back to standard form. The tool for that is completing the square — once for the x-terms, once for the y-terms.
🟢 Completing the Square — The Key Step
For any expression \(x^2 + bx\), the number to add is:
$$\left(\frac{b}{2}\right)^2$$
This turns \(x^2 + bx\) into a perfect square trinomial \(\left(x + \frac{b}{2}\right)^2\).
Important: whatever you add to one side, you must add to the other side too.
✏️ Worked Example
Convert to standard form: \(x^2 + y^2 + 6x – 4y – 12 = 0\)
👁️ Show Solution
Step 1 — Group x and y terms, move constant to the right:
$$(x^2 + 6x) + (y^2 – 4y) = 12$$
Step 2 — Complete the square for x:
Half of \(6\) is \(3\), and \(3^2 = 9\). Add \(9\) to both sides.
$$(x^2 + 6x + 9) + (y^2 – 4y) = 12 + 9$$
Step 3 — Complete the square for y:
Half of \(-4\) is \(-2\), and \((-2)^2 = 4\). Add \(4\) to both sides.
$$(x^2 + 6x + 9) + (y^2 – 4y + 4) = 12 + 9 + 4$$
Step 4 — Factor and simplify:
$$(x+3)^2 + (y-2)^2 = 25$$
Result: Center \((-3, 2)\), radius \(r = 5\)
Practice 1. Convert \(x^2 + y^2 – 4x + 8y + 4 = 0\) to standard form.
👁️ Show Solution
Group: \((x^2 – 4x) + (y^2 + 8y) = -4\)
Complete x: \(\left(\frac{-4}{2}\right)^2 = 4\) → add 4
Complete y: \(\left(\frac{8}{2}\right)^2 = 16\) → add 16
$$(x-2)^2 + (y+4)^2 = -4 + 4 + 16 = 16$$
Center: \((2, -4)\), radius \(r = 4\)
Practice 2. Convert \(x^2 + y^2 – 6x – 2y – 6 = 0\) to standard form.
👁️ Show Solution
Group: \((x^2 – 6x) + (y^2 – 2y) = 6\)
Complete x: \(\left(\frac{-6}{2}\right)^2 = 9\) → add 9
Complete y: \(\left(\frac{-2}{2}\right)^2 = 1\) → add 1
$$(x-3)^2 + (y-1)^2 = 6 + 9 + 1 = 16$$
Center: \((3, 1)\), radius \(r = 4\)
▶ Completing the Square with Fractions — Circles & Completing the Square | Get The Math
