Question

How do you write the equation of the parabola in vertex form given the vertex $\left( { - 2, - 8} \right)$and passes through the point $\left( {7, - 494} \right)$?

Answer 1

To solve this question, we need to use the standard formula for the vortex form of parabola. In this equation, we will first put the value of the given vortex point. After that we will put the value of the coordinates of the point through which the parabola is passing which will give us the value of the distance of the origin from the focus. Thus, we will finally get the required equation.

Formula used:
$y = a{\left( {x - h} \right)^2} + k$, where, $\left( {h,k} \right)$are the coordinates of the vertex, $\left( {x,y} \right)$are the coordinates of any point through which the parabola is passing and $a$is the distance of the origin from the focus

Complete step by step answer:
We are given the vertex $\left( { - 2, - 8} \right)$, Therefore we will take $\left( {h,k} \right) = \left( { - 2, \- 8} \right)$.
We will now use the vertex equation formula of parabola.

$$y = a{\left( {x - h} \right)^2} + k \\\ \Rightarrow y = a{\left( {x - \left( { - 2} \right)} \right)^2} + \left( { - 8} \right) \\\ \Rightarrow y = a{\left( {x + 2} \right)^2} - 8 \\\$$

Now, to obtain the equation, we need to find the value of . We will find this by using the point through which the parabola is passing that is point $\left( {7, - 494} \right)$. We will put $\left( {x,y} \right) = \left( {7, - 494} \right)$.

$$\Rightarrow - 494 = a{\left( {7 + 2} \right)^2} - 8 \\\ \Rightarrow - 494 = 81a - 8 \\\ \Rightarrow a = - 6 \\\$$

Now we will put this value in the main equation.
$\Rightarrow y = - 6{\left( {x + 2} \right)^2} - 8$

Thus, the required equation of the parabola in the vertex form is $y = - 6{\left( {x + 2} \right)^2} - 8$.

Note:
In this question, we have determined the equation of the parabola in the vertex form. It is important to know that the vertex is a point where the parabola crosses its axis of symmetry. In other words, the vortex of a parabola is the highest or lowest point of the parabola, also known as the maximum or minimum of a parabola.