To understand how the graph of [tex]\( f(x) = 3(4)^{x-5} + \frac{2}{3} \)[/tex] relates to its parent function, we need to break down the components and transformations applied to the parent function.
1. Identify the Parent Function:
The parent function here is [tex]\( g(x) = 4^x \)[/tex].
2. Horizontal Shift:
The term [tex]\( x-5 \)[/tex] inside the exponent indicates a horizontal shift. Specifically, [tex]\( x-5 \)[/tex] means that the function is shifted to the right by 5 units. This is because replacing [tex]\( x \)[/tex] with [tex]\( x-5 \)[/tex] in the parent function corresponds to a rightward shift.
3. Vertical Stretch:
The coefficient 3 in front of the [tex]\( (4)^{x-5} \)[/tex] term signifies a vertical stretch by a factor of 3. This means that the function's values are multiplied by 3, making it taller.
4. Vertical Shift:
The [tex]\( +\frac{2}{3} \)[/tex] outside of the exponential function means the entire graph is shifted upward by [tex]\( \frac{2}{3} \)[/tex] units.
Given these transformations, we need to select the primary transformation based on the choices provided:
A. The parent function has been compressed.
B. The parent function has been stretched.
C. The parent function has been translated up.
D. The parent function has been translated to the right.
While choice C (vertical shift up) and B (vertical stretch) are true transformations, the best description offered in the choices is the horizontal shift.
Thus, the answer that correctly relates the graph of [tex]\( f(x) \)[/tex] to its parent function is:
D. The parent function has been translated to the right.
