Question

How do you write `y=3x^2 + x - 2` in vertex form?

Answer 1

Use the form:

`y=a(x-h)^2+k" [1]"`

And the formulas:

`a =a" [2]"`

`h=-b/(2a)" [3]"`

`k = y(h)" [4]"`

Explanation:

The given equation:

`y=3x^2 + x - 2`

is in the standard form:

`y = ax^2+bx+c`

where `a = 3`, `b=1`, and `c = -2`

We shall use formula [2] to substitute 3 for "a" into equation [1]:

`y=3(x-h)^2+k" [1.1]"`

Use formula [3] to compute the value of h:

`h = -1/(2(3))`

`h = -1/6`

Substitute `-1/6` for h into equation [1.1]:

`y=3(x-(-1/6))^2+k" [1.2]"`

Note: You can write the above as `x+1/6` but doing so can cause an error, when you are asked for the x coordinate of the vertex, therefore, I do not recommend it.

Use formula [4] to determine the value of k:

`k = y(h)`

`k = y(-1/6)`

`k = 3(-1/6)^2 + (-1/6) - 2`

`k = -25/12`

Substitute `-25/12` for k into equation [1.2]:

`y=3(x-(-1/6))^2-25/12" [1.3]"`

Equation [1.3] is the answer.