Question

Write an equation of the graph in its final position.
The graph of y = 5^x is translated 9 units to the left and then 4 units upward.

a. y=5 x+9 −4
b. y=5 x−9 +4
c. y=5 x−9 +4
d. y=5 x+9 +4

Answer 1

To find the equation of the graph of \( y = 5^x \) after the specified translations, we need to apply the transformations step by step.

1. **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:
\[
y = 5^{(x + 9)} = 5^x \cdot 5^9
\]

2. **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. \( y = 5^{x + 9} + 4 \)**.