The equation of a circle is given in standard form. Complete parts a through c to follow the step by step process to sketch the graph of the circle.
x squared plus left parenthesis y minus 6 right parenthesis squared equals 36


We are given the equation of a circle in standard form:  
\[ x^2 + (y - 6)^2 = 36 \]

### Step-by-Step Process

#### (a) **Identify the center and radius of the circle**

The standard form of the equation of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Where:
- \((h, k)\) is the center of the circle
- \(r\) is the radius

Now, comparing the given equation with the standard form:

\[
x^2 + (y - 6)^2 = 36
\]

We can see that:
- \( (x - 0)^2 \) means \(h = 0\), so the \(x\)-coordinate of the center is \(0\).
- \( (y - 6)^2 \) means \(k = 6\), so the \(y\)-coordinate of the center is \(6\).
- The radius squared \( r^2 = 36 \), so the radius \(r = \sqrt{36} = 6\).

Thus:
- The center of the circle is \((0, 6)\).
- The radius is \(6\).

#### (b) **Plot the center and sketch the circle**

To sketch the graph:
1. **Plot the center** at the point \((0, 6)\) on the coordinate plane.
2. **Draw the circle** with radius \(6\). This means the circle will extend:
   - 6 units to the right and left of the center along the x-axis (i.e., from \(-6\) to \(6\) on the x-axis).
   - 6 units up and down from the center along the y-axis (i.e., from \(y = 0\) to \(y = 12\)).

#### (c) **Check for intersections with the axes**

- **Intersection with the x-axis**:
  Set \(y = 0\) in the equation:
  \[
  x^2 + (0 - 6)^2 = 36
  \]
  \[
  x^2 + 36 = 36 \quad \Rightarrow \quad x^2 = 0 \quad \Rightarrow \quad x = 0
  \]
  The circle intersects the x-axis at \((0, 0)\).

- **Intersection with the y-axis**:
  Set \(x = 0\) in the equation:
  \[
  0^2 + (y - 6)^2 = 36 \quad \Rightarrow \quad (y - 6)^2 = 36
  \]
  Taking the square root:
  \[
  y - 6 = \pm 6 \quad \Rightarrow \quad y = 12 \quad \text{or} \quad y = 0
  \]
  The circle intersects the y-axis at two points: \((0, 0)\) and \((0, 12)\).

### Final Answer:
- The **center** is at \((0, 6)\).
- The **radius** is \(6\).
- The **intersections** with the axes are \((0, 0)\) and \((0, 12)\).

Would you like more details or have any questions?

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Here are five questions to expand your understanding:

1. What is the general formula for the equation of a circle in standard form?
2. How would you transform the equation of a circle if it is not in standard form?
3. What happens to the graph of the circle if we change the radius?
4. How do you find the intersections of a circle with a diagonal line?
5. Can a circle have no intersections with the x or y axes? Under what conditions?

**Tip**: To quickly find the radius of a circle from its equation, always look for the term on the right-hand side of the equation. The radius is the square root of that value!

The equation of a circle is given in standard form. Complete parts a through c to follow the step by step process to sketch the graph of the circle. x squared plus left parenthesis y minus 6 right parenthesis squared equals 36.

Learn how to sketch the graph of the circle given by the equation x^2 + (y - 6)^2 = 36. This problem covers key concepts such as identifying the center and radius, plotting on the coordinate plane, and finding axis intersections. Suitable for students in Grades 9-10.

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