Question

Question
Which of the following functions is equivalent to the function below?

f(x)=x2−6x+8

(1 point)
Responses

f(x)=(x−6)2−28
f of x is equal to open paren x minus 6 close paren squared minus 28

f(x)=(x−2)(x−4)
f of x is equal to open paren x minus 2 close paren times open paren x minus 4 close paren

f(x)=(x−3)(x−3)
f of x is equal to open paren x minus 3 close paren times open paren x minus 3 close paren

f(x)=(x−3)2−9

Answer 1

To find which of the given options is equivalent to the function \( f(x) = x^2 - 6x + 8 \), we can perform some algebraic manipulations. We can start by factoring or completing the square.

1. **Factoring**:
We look for two numbers that multiply to \( 8 \) (the constant term) and add to \(-6\) (the coefficient of \( x \)). The numbers \(-2\) and \(-4\) work because:
\[
-2 \cdot -4 = 8 \quad \text{and} \quad -2 + (-4) = -6
\]
Thus, we can factor the quadratic:
\[
f(x) = (x - 2)(x - 4)
\]

2. **Completing the square**:
We can rewrite the function in vertex form. Start with the original function:
\[
f(x) = x^2 - 6x + 8
\]
We take half of \(-6\), square it, and rearrange:
\[
f(x) = (x^2 - 6x + 9) - 9 + 8
\]
This simplifies to:
\[
f(x) = (x - 3)^2 - 1
\]

From the options given, we can see the following:

- \( f(x) = (x - 6)^2 - 28 \) is not equivalent.
- \( f(x) = (x - 2)(x - 4) \) is equivalent (as shown in our factoring step).
- \( f(x) = (x - 3)(x - 3) \) simplifies to \( (x - 3)^2 \), which does not equal the original function.
- \( f(x) = (x - 3)^2 - 9 \) is also not equivalent since it doesn't produce the original quadratic.

Thus, the correct answer is:

**f(x) = (x−2)(x−4)**.