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
$(x+3)^2 + (y - 4)^2 = 28$

b) sketch the graph of the circle. choose the correct graph
a
b
c
d

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: Identify circle center

The standard circle form is $(x-h)^2+(y-k)^2=r^2$, where $(h,k)$ is the center. For $(x+3)^2+(y-4)^2=28$, rewrite $x+3$ as $x-(-3)$, so center is $(-3, 4)$.

Step2: Calculate circle radius

From $r^2=28$, solve for $r$:
$r=\sqrt{28}=2\sqrt{7}\approx5.29$

Step3: Find x-intercepts (set $y=0$)

Substitute $y=0$ into the circle equation:
$$(x+3)^2+(0-4)^2=28$$
$$(x+3)^2+16=28$$
$$(x+3)^2=12$$
$$x+3=\pm\sqrt{12}=\pm2\sqrt{3}$$
$$x=-3\pm2\sqrt{3}$$

Step4: Find y-intercepts (set $x=0$)

Substitute $x=0$ into the circle equation:
$$(0+3)^2+(y-4)^2=28$$
$$9+(y-4)^2=28$$
$$(y-4)^2=19$$
$$y-4=\pm\sqrt{19}$$
$$y=4\pm\sqrt{19}$$

Step5: Verify correct graph

The circle with center $(-3,4)$ and radius $\approx5.29$ matches option B (center left of y-axis, above x-axis, size consistent with radius).

Answer:

b) B (the graph with center at $(-3,4)$ and radius $\approx5.29$)
c) A. The x-intercept(s) is/are $x=-3+2\sqrt{3}, -3-2\sqrt{3}$
The y-intercept(s) is/are $y=4+\sqrt{19}, 4-\sqrt{19}$