Question

question: 10
which of the following functions is equivalent to $f(x) = (x - 3)^2 + 2$?

a. $g(x) = x^2 - 6x + 9$

b. $g(x) = x^2 - 6x + 11$

c. $g(x) = x^2 - 9$

d. $g(x) = x^2 - 3x + 2$

Explanation:

Step1: Expand the square term

We know that \((a - b)^2 = a^2 - 2ab + b^2\). For \((x - 3)^2\), \(a = x\) and \(b = 3\), so \((x - 3)^2 = x^2 - 2\times x\times3 + 3^2 = x^2 - 6x + 9\).

Step2: Add the constant term

Now, \(f(x)=(x - 3)^2 + 2\), substitute the expanded form of \((x - 3)^2\) we got above: \(f(x)=x^2 - 6x + 9 + 2\).

Step3: Simplify the expression

Combine the constant terms: \(9 + 2 = 11\), so \(f(x)=x^2 - 6x + 11\), which is equivalent to \(g(x)=x^2 - 6x + 11\) (option B).

Answer:

B. \(g(x)=x^2 - 6x + 11\)