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

1 Answer
Jul 1, 2017

Please see the explanation.

Explanation:

Given: $y = {x}^{2} - 2 x - 9 \text{ [1]}$

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

$y = a {x}^{2} + b x + c \text{ [2]}$

where $a = 1 , b = - 2 , \mathmr{and} c = - 9$

The vertex form of this type of parabola is:

$y = a {\left(x - h\right)}^{2} + k \text{ [3]}$

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]:

$y = {\left(x - h\right)}^{2} + k \text{ [4]}$

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

$h = \frac{- b}{2 a}$

Substituting in the know values:

$h = \frac{- \left(- 2\right)}{2 \left(1\right)}$

$h = 1$

Substitute the value of h into equation [4]:

$y = {\left(x - 1\right)}^{2} + k \text{ [5]}$

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

$k = {1}^{2} - 2 \left(1\right) - 9$

$k = - 10$

Substitute the value of k into equation [5]:

$y = {\left(x - 1\right)}^{2} - 10 \text{ [6]}$

Equation [6] is the vertex form.