Question

two functions are given.
$f(x) = -3x^2 + 2x + 1$
$g(x) = -3(x - 2)^2 + 2(x - 2) + 1$
which transformation maps $f$ onto $g$?
a a translation 2 units up
b a translation 2 units down
c a translation 2 units to the left
d a translation 2 units to the right

Explanation:

Step1: Recall Function Translation Rule

For a function \( y = f(x) \), the transformation \( y = f(x - h) \) represents a horizontal translation. If \( h>0 \), it's a translation \( h \) units to the right; if \( h<0 \), it's \( |h| \) units to the left.

Step2: Compare \( f(x) \) and \( g(x) \)

Given \( f(x)=-3x^{2}+2x + 1 \) and \( g(x)=-3(x - 2)^{2}+2(x - 2)+1 \). Notice that \( g(x) \) is obtained by replacing \( x \) with \( x - 2 \) in \( f(x) \), i.e., \( g(x)=f(x - 2) \).
By the translation rule, when we have \( y = f(x - 2) \), it means a translation of 2 units to the right from \( y = f(x) \).

Answer:

D. a translation 2 units to the right