Question

What is the vertex form of `x = (2y +5 )^2 + 21`?

Answer 1

`x = 4(y - (-2.5))^2+ 21`

Explanation:

Given: `x = (2y +5 )^2 + 21`

Note: There is a quick way to do this but it is easy to confuse yourself so I will do it the following way.

Expand the square:

`x = 4y^2+20y+25+21`

`x = 4y^2+20y+46" [1]"`

This is the standard form

`x = ay^2 +by + c`

where `a = 4, b = 20 and c = 46`

The general vertex form is:

`x = a(y - k)^2+ h" [2]"`

We know that `a` in the vertex form is the same as `a` in the standard form:

`x = 4(y - k)^2+ h" [2.1]"`

To find the value of k, use the formula:

`k = -b/(2a)`

`k = -20/(2(4)) = -2.5`

`x = 4(y - (-2.5))^2+ h" [2.2]"`

To find h, evaluate equation [1] at `x = k = -2.5`

`h = 4(-2.5)^2+20(-2.5)+46`

`h = 21`

`x = 4(y - (-2.5))^2+ 21" [2.3]"`