Answer :
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 follow these steps:
1. Understand the transformation: Replacing [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex] in any function results in shifting the graph of the function horizontally. A replacement of [tex]\( x \)[/tex] with [tex]\( x + a \)[/tex] shifts the graph [tex]\( a \)[/tex] units to the left.
2. Apply the replacement:
[tex]\[ f(x) = |x - 3| + 2 \][/tex]
If we replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex], the new function will be:
[tex]\[ f(x + 2) = |(x + 2) - 3| + 2 \][/tex]
3. Simplify the expression:
Firstly, simplify the absolute value term:
[tex]\[ (x + 2) - 3 = x - 1 \][/tex]
So the function becomes:
[tex]\[ f(x + 2) = |x - 1| + 2 \][/tex]
4. Analyze the result: The new function [tex]\( f(x + 2) = |x - 1| + 2 \)[/tex] indicates that the initial function [tex]\( f(x) = |x - 3| + 2 \)[/tex] has been transformed by shifting it horizontally.
By replacing [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex], the effect on the graph is a horizontal shift of 2 units to the left.
Therefore, the correct answer is:
[tex]\[ \text{The graph is shifted 2 units left.} \][/tex]
1. Understand the transformation: Replacing [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex] in any function results in shifting the graph of the function horizontally. A replacement of [tex]\( x \)[/tex] with [tex]\( x + a \)[/tex] shifts the graph [tex]\( a \)[/tex] units to the left.
2. Apply the replacement:
[tex]\[ f(x) = |x - 3| + 2 \][/tex]
If we replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex], the new function will be:
[tex]\[ f(x + 2) = |(x + 2) - 3| + 2 \][/tex]
3. Simplify the expression:
Firstly, simplify the absolute value term:
[tex]\[ (x + 2) - 3 = x - 1 \][/tex]
So the function becomes:
[tex]\[ f(x + 2) = |x - 1| + 2 \][/tex]
4. Analyze the result: The new function [tex]\( f(x + 2) = |x - 1| + 2 \)[/tex] indicates that the initial function [tex]\( f(x) = |x - 3| + 2 \)[/tex] has been transformed by shifting it horizontally.
By replacing [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex], the effect on the graph is a horizontal shift of 2 units to the left.
Therefore, the correct answer is:
[tex]\[ \text{The graph is shifted 2 units left.} \][/tex]
