Question

What are the vertex, focus, and directrix of the parabola with the given equation? y=1/28(x-4)^2-5

and explain.

Answer 1

The answer is:

vertex: (4,-5); focus: (4,2); directrix: y=-12

Answer 2

Vertex - (4,-5)

Focus (4,2)

Directrix y=-12

Explanation:

The equation
`y=(1)/(28)(x-4)^2-5` of the parabola shows that its vertex is at point (4,-5). Multiply the equation by 28:

`28y=(x-4)^2-140\Rightarrow (x-4)^2=28y+140,\\ \\(x-4)^2=28(y+5).`

The parameter p of the parabola is

`2p=28\Rightarrow p=14.`

The coordinates of the focus will be

`\left(4,-5+(p)/(2)\right)=\left(4,-5+(14)/(2)\right)=\left(4,2\right).`

The directrix has the equation

`y=-5-(p)/(2)\Rightarrow y=-5-7,\ y=-12.`