Write an equation of the horizontal parabola with the given vertex and passing through the given point. Vertex at (- 5, - 4) and passing through (- 7, - 2).

Hint: Analyse the given. Substitute the coordinates of vertex in the general equation of Horizontal parabola, and you’ll get an equation. In that equation, substitute the coordinates of the point in which the vertex passes through and find out the value of a. Now, try to get the final answer.

Complete step by step answer:

As you know that general equation of a horizontal parabola is:

${\left( {y - k} \right)^2} = 4a\left( {x - h} \right){\text{ }}$

As Vertex is given to us as (- 5, - 4), therefore we have

${\left( {y - \left( { - 4} \right)} \right)^2} = 4a\left( {x - \left( { - 5} \right)} \right){\text{ }}$

$\Rightarrow {\left( {y + 4} \right)^2} = 4a\left( {x + 5} \right){\text{ }} - Equation(1){\text{ }}$

Now, putting (- 7, - 2) in Equation(1) we get,

${\left( { - 2 + 4} \right)^2}{\text{ }} = {\text{ }}4a\left( { - 7 + 5} \right)$

$\Rightarrow {\left( 2 \right)^2}{\text{ }} = {\text{ }}4a\left( { - 2} \right){\text{ }}$

$\Rightarrow 4 = - 8a$

$\Rightarrow a = - \dfrac{1}{2}{\text{ }} - Equation(2)$

Using Equation(2) in Equation(1)

$\Rightarrow {(y + 4)^2} = 4\left( { - \dfrac{1}{2}} \right)\left( {x + 5} \right)$

$\Rightarrow {(y + 4)^2} = - 2(x + 5)$ is the required Equation.

Note: For these kinds of questions we must remember the general equation of a horizontal parabola and then put vertex's coordinates in it to get the equation in only one variable 'a'. Now, Put the coordinates of the point given to be on the parabola and solve to get the value of 'a'. Put the value of 'a' in the equation to get the desired equation.