Question

Using the given information, give the vertex form equation of each parabola.

Answer 1

Answer:

The equation of parabola is given by : $(x-4) = \frac{-1}{3}(y+3)^{2}$

Step-by-step explanation:

Given that vertex and focus of parabola are

Vertex: (4,-3)

Focus:($\frac{47}{12}$,-3)

The general equation of parabola is given by.

$(x-h)^{2} = 4p(y-k)$, When x-componet of focus and Vertex is same  

$(x-h) = 4p(y-k)^{2}$, When y-componet of focus and Vertex is same

where Vertex: (h,k)

and p is distance between vertex and focus

The distance between two points is given by :

L=$\sqrt{(X2-X1)^{2}+(Y2-Y1)^{2}}$

For value of p:

p=$\sqrt{(X2-X1)^{2}+(Y2-Y1)^{2}}$

p=$\sqrt{(4-\frac{47}{12})^{2}+((-3)-(-3))^{2}}$

p=$\sqrt{(\frac{1}{12})^{2}}$

p=$\frac{1}{12}$ and p=$\frac{-1}{12}$

Since, Focus is left side of the vertex,

p=$\frac{-1}{12}$ is required value

Replacing value in general equation of parabola,

Vertex: (h,k)=(4,-3)

p=$\frac{-1}{12}$

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

$(x-4) = 4(\frac{-1}{12})(y+3)^{2}$

$(x-4) = \frac{-1}{3}(y+3)^{2}$