To determine the translation applied to get the graph of [tex]\( g(x) = x + 3 \)[/tex] from the graph of [tex]\( f(x) = x \)[/tex], we need to analyze how the two functions relate to each other.
1. Understand how translations work:
- A vertical translation affects the y-values of the function. If [tex]\( g(x) \)[/tex] is [tex]\( f(x) \)[/tex] shifted vertically, we get [tex]\( g(x) = f(x) + k \)[/tex], where [tex]\( k \)[/tex] is the number of units shifted up or down.
- A horizontal translation affects the x-values of the function. If [tex]\( g(x) \)[/tex] is [tex]\( f(x) \)[/tex] shifted horizontally, we get [tex]\( g(x) = f(x - h) \)[/tex], where [tex]\( h \)[/tex] is the number of units shifted left or right.
2. Analyze the given functions:
- Given [tex]\( f(x) = x \)[/tex] and [tex]\( g(x) = x + 3 \)[/tex].
3. Determine the type of translation:
- Notice that [tex]\( g(x) = x + 3 \)[/tex] can be rewritten as [tex]\( g(x) = f(x) + 3 \)[/tex].
Since the function [tex]\( g(x) \)[/tex] adds 3 to the x-value of [tex]\( f(x) = x \)[/tex] without changing the form of the function (the y-values), this indicates a shift in the x-direction.
4. Identify the direction of the shift:
- To shift [tex]\( f(x) \)[/tex] horizontally to get [tex]\( g(x) \)[/tex], we need to determine the value and the direction of the shift. We see [tex]\( g(x) = f(x) + 3 \)[/tex] means all values of [tex]\( f(x) \)[/tex] are effectively shifted.
- When considering the direction: [tex]\( g(x) = f(x) + 3 \)[/tex] shifts everything 3 units to the left.
Therefore, the correct translation is a horizontal translation of 3 units to the left.
Select from the drop-down menu:
a horizontal translation of 3 units to the left
