Question

What is this function written in vertex form?

Answer 1

Answer:

$f(x) = (x-1)^{2} -7$ is the correct answer.

Step-by-step explanation:

Given that function f(x) is:

$f(x) = x^{2} -2x-6$

f(x) is a quadratic function in x, meaning that it has a maximum power of 2 of x.

Vertex form of quadratic function is given as:

$f (x) = a(x - h)^2 + k$

i.e. we make whole square of terms of $x$.

Now, let us try to make whole square term of $x$.

$f(x) = x^{2} -2x-6$

Adding and subtracting 1 from RHS:

$f(x) = x^{2} -2x-6+1-1\\f(x) = (x^{2} -2x+1)-1-6\\f(x) = (x^{2} -2\times x\times 1+1^2)-7$

Now, using the formula:

$(a-b)^2 = a^2 -2ab+b^2$

The given function becomes:

$f(x) = (x-1)^{2}-7$

It is comparable to vertex form i.e. $f (x) = a(x - h)^2 + k$

where a = 1, h = 1 and k = -7

Hence, the vertex form of given function is:

$f(x) = (x-1)^{2}-7$