The vertex of this parabola is at (2,-4). When the y-value is -3, the x-value -3. What is the coefficient of the squared term in the parabola's equation?

Question

4666 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-23T18:07:36

There are two parabolas that meet your specifications.

The first parabola is of the form:

$y = a_{1} {(x - 2)}^{2} - 4$ $y = a_1(x-2)^2-4$

The second parabola is of the form:

$x = a_{2} {(y - - 4)}^{2} + 2$ $x = a_2(y- -4)^2+2$

Find the value of $a_{1}$ $a_1$ by substituting $y (- 3) = - 3$ $y(-3) = -3$ :

$- 3 = a_{1} {(- 3 - 2)}^{2} - 4$ $-3 = a_1(-3-2)^2-4$

$1 = a_{1} {(- 3 - 2)}^{2}$ $1 = a_1(-3-2)^2$

$a_{1} = \frac{1}{25}$ $a_1 = 1/25$

Find the value of $a_{2}$ $a_2$ by substituting $x (- 3) = - 3$ $x(-3) = -3$ :

$- 3 = a_{2} {(- 3 - - 4)}^{2} + 2$ $-3 = a_2(-3- -4)^2+2$

$a_{2} = - 5$ $a_2 = -5$

The two equations that meet your requirements are:

$y = \frac{1}{25} {(x - 2)}^{2} - 4$ $color(red)(y = 1/25(x-2)^2-4)$ and $x = - 5 {(y - - 4)}^{2} + 2$ $color(blue)(x = -5(y- -4)^2+2)$

Please observe that both parabolas have their vertex at $(2, - 4)$ $(2,-4)$ and contain the point $(- 3, - 3)$ $(-3,-3)$ :

![www.desmos.com/calculator]( useruploads.socratic.org )