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Question 9 of 10

How does the graph of [tex]f(x)=3(4)^{x-5}+\frac{2}{3}[/tex] relate to its parent function?

A. The parent function has been compressed.
B. The parent function has been translated up.
C. The parent function has been stretched.
D. The parent function has been translated to the right.

Sagot :

GinnyAnswer
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.
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