The equation of a circle is given below. Identify the center and the radius. Then graph the circle. $$ x^{2}+y^{2}-6 x+8 y=-9 $$ $$ \text { Center: }(\square, \square) $$

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We start with the equation of the circle:

$$
x^{2}+y^{2}-6x+8y=-9
$$

**Step 1: Group the $$x$$ terms and $$y$$ terms.**

$$
(x^2 - 6x) + (y^2 + 8y) = -9
$$

**Step 2: Complete the square for the $$x$$ terms.**

For the expression $$x^2 - 6x$$:
- Take half of the coefficient of $$x$$: $$-\frac{6}{2} = -3$$.
- Square it: $$(-3)^2 = 9$$.

Add and subtract $$9$$:

$$
x^2 - 6x + 9 - 9
$$

This gives $$(x-3)^2 - 9$$.

**Step 3: Complete the square for the $$y$$ terms.**

For the expression $$y^2 + 8y$$:
- Take half of the coefficient of $$y$$: $$\frac{8}{2} = 4$$.
- Square it: $$4^2 = 16$$.

Add and subtract $$16$$:

$$
y^2 + 8y + 16 - 16
$$

This gives $$(y+4)^2 - 16$$.

**Step 4: Substitute the completed squares into the equation.**

$$
(x-3)^2 - 9 + (y+4)^2 - 16 = -9
$$

Combine the constant terms:

$$
(x-3)^2 + (y+4)^2 - 25 = -9
$$

**Step 5: Isolate the circle equation in standard form.**

Add $$25$$ to both sides:

$$
(x-3)^2 + (y+4)^2 = 16
$$

**Step 6: Identify the center and radius.**

The standard form of a circle is:

$$
(x-h)^2 + (y-k)^2 = r^2
$$

Comparing, we have:

- $$h = 3$$
- $$k = -4$$
- $$r^2 = 16$$, hence $$r = 4$$.

$$
\text{Center: } (3, -4) \quad \text{and} \quad \text{Radius: } 4
$$

**Step 7: Graphing the Circle**

1. **Plot the center** at $$(3, -4)$$.
2. **Draw a circle** with radius $$4$$ around the center. This includes all points that are a distance of $$4$$ units from $$(3, -4)$$.

The circle will touch points:
- To the right: $$(3+4, -4) = (7, -4)$$
- To the left: $$(3-4, -4) = (-1, -4)$$
- Upward: $$(3, -4+4) = (3, 0)$$
- Downward: $$(3, -4-4) = (3, -8)$$
Center: (3, -4), Radius: 4

The equation of a circle is given below. Identify the center and the radius. Then graph the circle.

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