To determine how the graph of the function [tex]\( f(x) = 3(4)^{x-5} + \frac{2}{3} \)[/tex] relates to its parent function [tex]\( g(x) = 4^x \)[/tex], we need to analyze the transformations that have been applied.
Let's examine the function step-by-step:
1. Horizontal Shift:
- The term [tex]\( (4)^{x-5} \)[/tex] indicates a horizontal shift. The expression [tex]\( x-5 \)[/tex] inside the exponent means the graph is shifted to the right by 5 units. This is because, for any input [tex]\( x \)[/tex], we effectively have to add 5 to it to get the same result as the parent function would have without the shift.
2. Vertical Stretch:
- The coefficient 3 in [tex]\( 3(4)^{x-5} \)[/tex] represents a vertical stretch by a factor of 3. This means that each output value of the parent function has been multiplied by 3.
3. Vertical Shift:
- The term [tex]\( +\frac{2}{3} \)[/tex] represents a vertical shift upwards by [tex]\(\frac{2}{3}\)[/tex] units. This means that after applying the horizontal shift and vertical stretch, we add [tex]\(\frac{2}{3}\)[/tex] to each function value.
Given these transformations:
- A horizontal shift to the right by 5 units.
- A vertical stretch by a factor of 3.
- A vertical shift upward by [tex]\(\frac{2}{3}\)[/tex].
Now, let's examine the options:
A. The parent function has been compressed.
- This option is incorrect because the function actually experiences a vertical stretch, not a compression.
B. The parent function has been translated up.
- This option is partially correct because there is a vertical shift upward by [tex]\(\frac{2}{3}\)[/tex], but it doesn't account for all the transformations.
C. The parent function has been stretched.
- This option is also partially correct because there is a vertical stretch by a factor of 3. However, it doesn't consider the horizontal shift.
D. The parent function has been translated to the right.
- This option is correct and the most comprehensive, as it considers the dominant horizontal shift to the right by 5 units.
Considering the horizontal shift is a key transformation in this context, option D is the best answer:
D. The parent function has been translated to the right.
