To determine the effect of replacing [tex]\( x \)[/tex] with [tex]\( x+2 \)[/tex] on the graph of the function [tex]\( f(x) = |x - 3| + 2 \)[/tex], let's go through the changes step by step.
1. Original Function:
The original function is [tex]\( f(x) = |x - 3| + 2 \)[/tex].
2. Replacing [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex]:
The new function becomes [tex]\( f(x + 2) \)[/tex]. Let's substitute [tex]\( x + 2 \)[/tex] into the original function:
[tex]\[
f(x + 2) = |(x + 2) - 3| + 2
\][/tex]
Simplify the expression inside the absolute value:
[tex]\[
f(x + 2) = |x + 2 - 3| + 2
\][/tex]
[tex]\[
f(x + 2) = |x - 1| + 2
\][/tex]
3. Analysis of the Change:
Compare this new function [tex]\( f(x + 2) = |x - 1| + 2 \)[/tex] with the original function [tex]\( f(x) = |x - 3| + 2 \)[/tex]:
- The expression inside the absolute value has changed from [tex]\( |x - 3| \)[/tex] to [tex]\( |x - 1| \)[/tex].
- When you replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex], effectively we are shifting the graph of the function [tex]\( 2 \)[/tex] units to the left.
Therefore, the effect of replacing [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex] in the function [tex]\( f(x) \)[/tex] results in shifting the graph 2 units left. The correct answer is:
- The graph is shifted 2 units left.
