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