Question

Write the equation of a parabola having the vertex (1, −2) and containing the point (3, 6) in vertex form. Then, rewrite the equation in standard form. [Hint: Vertex form: y - k = a(x - h)2]

Answer 1

PART A

The equation of the parabola in vertex form is given by the formula,

$y - k = a {(x - h)}^{2}$

where

$(h,k)=(1,-2)$

is the vertex of the parabola.

We substitute these values to obtain,


$y + 2 = a {(x - 1)}^{2}$

The point, (3,6) lies on the parabola.

It must therefore satisfy its equation.


$6 + 2 = a {(3 - 1)}^{2}$


$8= a {(2)}^{2}$


$8=4a$


$a = 2$
Hence the equation of the parabola in vertex form is


$y + 2 = 2 {(x - 1)}^{2}$


PART B

To obtain the equation of the parabola in standard form, we expand the vertex form of the equation.

$y + 2 = 2{(x - 1)}^{2}$

This implies that

$y + 2 = 2(x - 1)(x - 1)$


We expand to obtain,


$y + 2 = 2( {x}^{2} - 2x + 1)$


This will give us,


$y + 2 = 2 {x}^{2} - 4x + 2$


$y = {x}^{2} - 4x$

This equation is now in the form,

$y = a {x}^{2} + bx + c$
where

$a=1,b=-4,c=0$

This is the standard form