Question

a) find the center - radius form of the equation of the circle with center (-1,3) and radius 4. b) graph the circle. a) the center - radius form of the equation of the circle is (type an equation.) b) use the graphing tool to graph the circle. click to enlarge graph

Explanation:

Step1: Recall circle - equation formula

The center - radius form of a circle's equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius.

Step2: Identify values of $h$, $k$, and $r$

Given center $(-1,3)$ and radius $r = 4$, so $h=-1$, $k = 3$, $r=4$.

Step3: Substitute values into formula

Substitute $h=-1$, $k = 3$, and $r = 4$ into the formula: $(x-(-1))^2+(y - 3)^2=4^2$, which simplifies to $(x + 1)^2+(y - 3)^2=16$.

Answer:

a) $(x + 1)^2+(y - 3)^2=16$
b) To graph the circle:

1. Plot the center point $(-1,3)$ on the coordinate plane.
2. From the center, move 4 units up, down, left, and right to get four points on the circle: $(-1,3 + 4)=(-1,7)$, $(-1,3-4)=(-1,-1)$, $(-1+4,3)=(3,3)$, $(-1 - 4,3)=(-5,3)$.
3. Sketch the circle passing through these four points. Since this is a text - based response, the actual graphing using a tool (as required in the problem) would be done on graph paper or a graphing utility like Desmos, GeoGebra etc. by entering the equation $(x + 1)^2+(y - 3)^2=16$.