Question

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.
$x^{2}+(y - 4)^{2}=16$
the radius is 4
(simplify your answer. type an exact answer, using radicals as needed.)

b) sketch the graph of the circle.
use the graphing tool to graph the circle.

c) determine the x - and y - intercepts of the circle. select the correct choice and, if necessary, fill in the answer box within your choice.

a. the x - intercept(s) is/are x =
(use a comma to separate answers as needed. type an exact answer using radicals as needed.)

b. there is no x - intercept

Explanation:

Step1: Recall circle - intercept formulas

The standard form of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. For the given circle $x^{2}+(y - 4)^{2}=16$, the center is $(0,4)$ and $r = 4$.
To find the $x$-intercepts, set $y = 0$:
$x^{2}+(0 - 4)^{2}=16$

Step2: Solve for $x$

$x^{2}+16 = 16$
$x^{2}=16 - 16$
$x^{2}=0$
$x = 0$
To find the $y$-intercepts, set $x = 0$:
$(0)^{2}+(y - 4)^{2}=16$
$(y - 4)^{2}=16$
$y-4=\pm4$
When $y - 4 = 4$, $y=8$; when $y - 4=-4$, $y = 0$.

Step3: Graph the circle

The center of the circle is at $(0,4)$ with a radius of 4. Plot the center, then use the radius to find points on the circle.

Answer:

b) (Graphing a circle by hand or using a graph - ing utility: Plot the center at $(0,4)$ and then mark points 4 units away from the center in all directions to draw the circle.)
c)
A. The $x$-intercept(s) is/are $x = 0$
The $y$-intercept(s) is/are $y=0,8$