Question

How do you write `y=x^2-2x-9` in vertex form?

Answer 1

Please see the explanation.

Explanation:

Given: `y=x^2-2x-9" [1]"`

We observe that equation [1] is in the standard form:

`y = ax^2+bx+c" [2]"`

where `a = 1, b = -2, and c = -9`

The vertex form of this type of parabola is:

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

The "a" in equation [2] and the "a" in equation [3] are the same attribute of a parabola, therefore, we may substitute 1 for "a" into equation [3]:

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

We know that "h" is the x coordinate of the axis of the vertex given by the formula:

`h = (-b)/(2a)`

Substituting in the know values:

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

`h = 1`

Substitute the value of h into equation [4]:

`y = (x-1)^2+k" [5]"`

We know that k is the y coordinate of the vertex. We can find the value of k by evaluating the function at h:

`k = 1^2-2(1) -9`

`k = -10`

Substitute the value of k into equation [5]:

`y = (x-1)^2-10" [6]"`

Equation [6] is the vertex form.