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Question 8(Multiple Choice Worth 3 points)
(04.04 MC)

Given the directrix of y = 4 and focus of (0, 2), which is the equation of the parabola?

y = one fourthx2 + 3
y = −one fourthx2 + 3
y = −one fourthx2 − 3
y = one fourthx2 − 3

Respuesta :

calculista

Answer:

[tex]y=-\frac{1}{4}x^2+3[/tex]

y = −one fourthx2 + 3

Step-by-step explanation:

we know that

The standard form of the equation of the vertical parabola is equal to

[tex](x-h)^2=4p(y-k)[/tex]

where

(h,k) is the vertex

p is the focal distance (Distance from the vertex to the focus)

Remember that the distance from the vertex to the focus is the same that the distance from the vertex to the directrix

In this problem we have

Directrix y=4

Focus (0,2)

Is a vertical parabola open downward

The distance from the focus to the directrix is equal to (4-2) =2 units

so

The distance from the vertex to the focus is half the distance from the focus to the directrix

therefore

p=1

The x-coordinate of the vertex is the same that the  x-coordinate of the focus

The y-coordinate of the vertex is equal to the y-coordinate of the focus plus the focal distance p

so

The vertex is the point (0,2+1) ----> (0,3)

substitute the given values

[tex](x-0)^2=-4(1)(y-3)[/tex] ----> is negative because the parabola open downward

[tex]x^2=-4(y-3)[/tex]

isolate the variable y

[tex]y=-\frac{1}{4}x^2+3[/tex]

Answer:

y = −one fourthx2 + 3

Step-by-step explanation:

we know that

The standard form of the equation of the vertical parabola is equal to

where

(h,k) is the vertex

p is the focal distance (Distance from the vertex to the focus)

Remember that the distance from the vertex to the focus is the same that the distance from the vertex to the directrix

In this problem we have

Directxy=4

Focus (0,2)

Is a vertical parabola open downward

The distance from the focus to the directrix is equal to (4-2) =2 units

so

The distance from the vertex to the focus is half the distance from the focus to the directrix

therefore

p=1

The x-coordinate of the vertex is the same that the  x-coordinate of the focus

The y-coordinate of the vertex is equal to the y-coordinate of the focus plus the focal distance p

so

The vertex is the point (0,2+1) ----> (0,3)

substitute the given values

----> is negative because the parabola open downward

isolate the variable y

Step-by-step explanation:

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