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: $$-2\cdot -4=8\text{and}-2+(-4)=-6$$ 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).