5. determine the radius and center of the following circle and sketch the graph of the circle.\n$$(x+1)^{2}+(…

Explanation:

Step1: Identify standard circle equation form

The standard form is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Step2: Extract center coordinates $(h, k)$

Comparing $(x + 1)^2 + (y + 3)^2 = 6$ to standard form, $h = -1$ and $k = -3$.

Step3: Calculate the radius $r$

Set $r^2 = 6$, then solve for $r$ by taking the square root. $$r = \sqrt{6} \approx 2.45$$

Step4: Describe the graph sketch

Plot center $(-1, -3)$ and draw a circle with radius $\sqrt{6}$ units.

Answer:

Center: $(-1, -3)$ Radius: $\sqrt{6}$ Graph: A circle centered at $(-1, -3)$ passing through points approximately $2.45$ units away in all directions.