Question

The equation of a parabola is given. y=−14x2+4x−19 What are the coordinates of the vertex of the parabola?

Answer 1

The coordinates for the vertex of the parabola is:
Minimum ( -1/7, -135/7)

Answer 2

We know that the equation of a parabola is as:

`y=a(x-h)^2+k`................(Equation 1)

Where h and k are the coordinates of the vertex of the parabola which is at
`(h,k)`.

Now, The equation of the parabola given to us is:

`y=-14x^2+4x-19`..........(Equation 2)

Thus to find the vertex of this parabola, we will have to first convert (Equation 2) to (Equation 1) and find the corresponding value of h and k. For this we will have to complete the square and balance the equation. Let us begin:

`y=-14x^2+4x-19`..........(Equation 2)

`y=-14(x^2-(4)/(14)x)-19=-14(x^2+(2)/(7)x)-19`

`\therefore y=-14(x^2-2* (1)/(7)* x)-19`

`y=-14(x^2-2* (1)/(7)* x+(1)/(49)-(1)/(49))-19`

`y=-14(x^2-(2)/(7)x+(1)/(49))+(14)/(49) -19`

`y=-14(x-(1)/(7))^2-(131)/(7)`

Thus,
`h=(1)/(7)` and

`k=(-131)/(7)`

Therefore, the coordinates of the vertex of the parabola are:
`((1)/(7),(-131)/(7) )`