To understand how the graph of [tex]\( f(x) = 3 \cdot 4^{x-5} + \frac{2}{3} \)[/tex] relates to its parent function, we need to analyze the transformations in detail.
1. Identify the parent function:
The parent function here is [tex]\( f(x) = 4^x \)[/tex].
2. Horizontal Translation:
The term [tex]\( (x-5) \)[/tex] inside the exponent suggests a horizontal shift. Typically, [tex]\( f(x - h) \)[/tex] translates the graph of [tex]\( f(x) \)[/tex] horizontally to the right by [tex]\( h \)[/tex] units. Hence, [tex]\( 4^{x-5} \)[/tex] means the graph is translated to the right by 5 units.
3. Vertical Stretch:
The coefficient 3 outside the base term [tex]\( 4^{x-5} \)[/tex] indicates a vertical stretch. Specifically, multiplying the parent function [tex]\( f(x) \)[/tex] by a factor of 3 stretches it vertically by a factor of 3.
4. Vertical Translation:
The constant term [tex]\( \frac{2}{3} \)[/tex] added at the end of the function indicates a vertical translation. Adding [tex]\( \frac{2}{3} \)[/tex] to the function translates the graph up by [tex]\( \frac{2}{3} \)[/tex] units.
Given these transformations, the most prominent transformation is:
- The translation to the right by 5 units, which is a horizontal translation affecting the position of the entire graph.
Therefore, the correct answer to the question regarding the relation to its parent function is:
C. The parent function has been translated to the right.
