Question

identify the choice that best completes the statement or answers the question.
find the vertex and focus of the parabola.
$(y - 2)^2+16(x - 3)=0$
select one:
a. vertex $(-3,-2)$ focus: $(-3,14)$
b. vertex $(-3,-2)$ focus: $(-7,-2)$
c. vertex $(3,2)$ focus: $(3,-2)$
d. vertex $(3,2)$ focus: $(-1,2)$
e. vertex $(-3,-2)$ focus: $(-3,-18)$

Explanation:

Step1: Rewrite the equation

Rewrite \((y - 2)^2+16(x - 3)=0\) as \((y - 2)^2=-16(x - 3)\). The standard form of a parabola that opens to the left is \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex. Here \(h = 3,k = 2\), so the vertex is \((3,2)\).

Step2: Find the value of \(p\)

Compare \((y - 2)^2=-16(x - 3)\) with \((y - k)^2 = 4p(x - h)\). We have \(4p=-16\), so \(p=- 4\).

Step3: Find the focus

For a parabola of the form \((y - k)^2 = 4p(x - h)\) (opens left or right), the focus is \((h + p,k)\). Substitute \(h = 3,k = 2,p=-4\) into \((h + p,k)\), we get \((3+(-4),2)=(-1,2)\)

Answer:

D. vertex \((3,2)\) focus: \((-1,2)\)