Question

The equation of a parabola is given. y=1/2 x 2+6x+24 What is the equation of the directrix of the parabola? Enter your answer in the box. will report if u give some random answer just to get points

Answer 1

Answer: y = 5.5

Step-by-step explanation:

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

Step 1: find the vertex

axis of symmetry: $x = \dfrac{-b}{2a} = \dfrac{-6}{2(\frac{1}{2})} =\dfrac{-6}{1} = -6$

y-coordinate of vertex: $y=\dfrac{1}{2}(-6)^2+6(-6)+24$

= 18 - 36 + 24

= 6

Vertex: (-6, 6)

Step 2: find p (the distance from the vertex to the focus): $\dfrac{1}{4p} =a$

$\dfrac{1}{4p} =\dfrac{1}{2}$

cross multiply to get: 2 = 4p

divide both sides by 4 to get: $p =\dfrac{1}{2}$

Step 3: find the directrix

Since the parabola opens up, the focus will be above the vertex.

focus: (-6, 6 + $\dfrac{1}{2}$) = (-6, 6.5)

and the directrix will be below the vertex.

directrix: y = 6 - $\dfrac{1}{2}$ ⇒ y = 5.5

Answer 2

The directrix also gets shiftd toward right.  Then the equation of the directrix of the parabola will be y = 11/2.

What is the parabola?

The equation of a parabola function is given as,

y² = 4ax

x² = 4ay

Where a is the coordinate of the focus.

The equation of a parabola is given.

y = (1/2) x² + 6x + 24

Simplify the equation, we have

2y = x² + 12x + 48

2y = x² + 12x + 36 + 12

2y = (x + 6)² + 12

Then the equation can be written as,

4(1/2)y = (x + 6)² + 12

Compare with standard equation, we have

a = (-1/2, 0)

The parabola is gets shifted toward right.

Then the directrix also gets shiftd toward right.

Then the equation the directrix will be

y = -1/2 + 6

y = 11/2

More about the parabola link is given below.

https://brainly.com/question/8495504

#SPJ2