Question

The equation of a parabola is given.

y = ½x² + 6x + 24

What is the equation of the directrix of the parabola?

Enter your answer in the box.

_________

Answer 1

Answer:

The equation of the directrix of the parabola is $y=5.5$

Step-by-step explanation:

we know that

The equation of a vertical parabola in standard form is equal to

$(x-h)^{2}=4a(y-k)$

where

(h,k) is the vertex of the parabola

y=k-a is the directrix of the parabola

In this problem we have

$y=\frac{1}{2}x^{2}+6x+24$

Convert to standard form

$y-24=\frac{1}{2}x^{2}+6x$

$y-24=\frac{1}{2}(x^{2}+12x)$

$y-24+18=\frac{1}{2}(x^{2}+12x+36)$

$y-6=\frac{1}{2}(x^{2}+12x+36)$

$y-6=\frac{1}{2}(x+6)^{2}$

therefore

$(x+6)^{2}=2(y-6)$ ----> standard form

The vertex is the point (-6,6)

$4a=2$

$a=\frac{1}{2}=0.5$

The  directrix of the parabola is

$y=k-a$

$y=6-0.5=5.5$

therefore

The equation of the directrix of the parabola is $y=5.5$

Answer 2

Answer: y = 5.5

Just comfirm that the answer above is correct :)