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)

$y=-14(x^2-\frac{4}{14}x)-19=-14(x^2+\frac{2}{7}x)-19$

$\therefore y=-14(x^2-2\times \frac{1}{7}\times x)-19$

$y=-14(x^2-2\times \frac{1}{7}\times x+\frac{1}{49}-\frac{1}{49})-19$

$y=-14(x^2-\frac{2}{7}x+\frac{1}{49})+\frac{14}{49} -19$

$y=-14(x-\frac{1}{7})^2-\frac{131}{7}$

Thus, $h=\frac{1}{7}$ and

$k=\frac{-131}{7}$

Therefore, the coordinates of the vertex of the parabola are: $(\frac{1}{7},\frac{-131}{7} )$