To find the equation of the graph of \( y = 5^x \) after the specified translations, we need to apply the transformations step by step.
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Translation 9 units to the left: This translation affects the \( x \)-coordinate. The transformation rule for translating a function \( f(x) \) left by \( h \) units is given by \( f(x + h) \). Therefore, translating \( y = 5^x \) to the left by 9 units gives: \[ y = 5^{(x + 9)} = 5^x \cdot 5^9 \]
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Translation 4 units upward: This translation affects the \( y \)-coordinate. The transformation rule for moving a function \( f(x) \) up by \( k \) units is \( f(x) + k \). Thus, translating \( 5^{(x + 9)} \) upward by 4 units gives: \[ y = 5^{(x + 9)} + 4 \]
Putting it all together, the final equation of the transformed graph is: \[ y = 5^{(x + 9)} + 4 \]
Now identifying the correct corresponding choice from the options provided:
a. \( y = 5^{x + 9} - 4 \)
b. \( y = 5^{x - 9} + 4 \)
c. \( y = 5^{x - 9} + 4 \)
d. \( y = 5^{x + 9} + 4 \)
The correct answer is: d. \( y = 5^{x + 9} + 4 \).