Question

homework assignment 8.3 the parabola
score: 6.35/13 answered: 9/13
question 10
find the coordinates of the vertex and the focus for the parabola given by the following equation:
$x^{2}-4x - 36y-32 = 0$
round all answers to 2 places after the decimal point, if necessary.
coordinates of the vertex: ( , )
coordinates of the focus: ( , )
question help: video written example

Explanation:

Step1: Rearrange equation

Isolate y terms: $x^2 - 4x = 36y + 32$

Step2: Complete the square for x

$x^2 -4x +4 = 36y +32 +4$ → $(x-2)^2 = 36y +36$

Step3: Simplify to vertex form

$(x-2)^2 = 36(y +1)$ → $(x-2)^2 = 4*9(y - (-1))$

Step4: Find vertex

Vertex $(h,k)=(2, -1)$

Step5: Find focus

$4p=36$ → $p=9$; focus $(h, k+p)=(2, -1+9)=(2,8)$

Answer:

Coordinates of the vertex: (2, -1)
Coordinates of the focus: (2, 8)