Give the center and radius of the circle described by the
equation and graph the equation. Use the graph to identify
the domain and range.

Question

Give the center and radius of the circle described by the equation and graph the equation. Use the graph to identify the domain and range. $$ (x+5)^{2}+y^{2}=25 $$

UpStudy AI · Accepted Answer

**Step 1: Identify the center and radius of the circle**

The given equation is 
$$
(x+5)^2 + y^2 = 25.
$$
This is in the standard form for a circle:
$$
(x-h)^2 + (y-k)^2 = r^2,
$$
where $$(h,k)$$ is the center and $$r$$ is the radius.

In our equation:
- The term $$(x+5)^2$$ can be written as $$(x-(-5))^2$$. Thus, $$h = -5$$.
- The term $$y^2$$ is $$(y-0)^2$$. Thus, $$k = 0$$.

Also, we have 
$$
r^2 = 25 \implies r = 5.
$$

The center of the circle is $$(-5, 0)$$ and the radius is $$5$$.

**Step 2: Graph the circle**

To sketch the circle:
- Plot the center at $$(-5, 0)$$.
- From the center, move $$5$$ units in all directions to form the circle. The points on the circle include:
  - $$(-5+5,0) = (0,0)$$ on the right,
  - $$(-5-5,0) = (-10,0)$$ on the left,
  - $$(-5,0+5) = (-5,5)$$ upward,
  - $$(-5,0-5) = (-5,-5)$$ downward.

**Step 3: Determine the domain and range**

The circle is defined for all points $$(x,y)$$ that satisfy the circle equation.

- The $$x$$-coordinate ranges from the leftmost point $$-10$$ to the rightmost point $$0$$. Thus, the domain is 
$$
[-10, 0].
$$

- The $$y$$-coordinate ranges from the bottom point $$-5$$ to the top point $$5$$. Thus, the range is 
$$
[-5, 5].
$$

**Summary**

- **Center:** $$(-5, 0)$$
- **Radius:** $$5$$
- **Domain:** $$[-10, 0]$$
- **Range:** $$[-5, 5]$$

A graph of the circle would show a circle centered at $$(-5, 0)$$ with a radius of $$5$$, touching the points $$(0,0)$$, $$(-10,0)$$, $$(-5,5)$$, and $$(-5,-5)$$.
The circle has a center at $$(-5, 0)$$ and a radius of $$5$$. Its domain is $$[-10, 0]$$ and its range is $$[-5, 5]$$.

Give the center and radius of the circle described by the
equation and graph the equation. Use the graph to identify
the domain and range.

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