Question
Which functions shows the function \( f(x)=3^{x} \) being translated 5 units down? \( f(x)=3^{x}+5 \) \( f(x)=3^{x}-5 \) \( f(x)=3^{x+5} \) \( f(x)=3^{x-5} \)
Ask by Gray Higgins. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(x) = 3^{x} - 5 \)
Solution
To translate the function \( f(x) = 3^x \) **5 units down**, you subtract 5 from the original function. This results in:
\[
f(x) = 3^x - 5
\]
Let's briefly analyze why this is the correct transformation:
- **Upward/Downward Shifts:** Adding a constant shifts the graph **up**, while subtracting a constant shifts it **down**.
- **Horizontal Shifts:** Changing the input inside the exponent (like \( x + 5 \) or \( x - 5 \)) shifts the graph **left** or **right**, respectively.
Given the options:
1. \( f(x) = 3^x + 5 \) shifts the graph **up** by 5 units.
2. \( f(x) = 3^x - 5 \) shifts the graph **down** by 5 units.
3. \( f(x) = 3^{x+5} \) shifts the graph **left** by 5 units.
4. \( f(x) = 3^{x-5} \) shifts the graph **right** by 5 units.
**Therefore, the correct function that represents \( f(x) = 3^x \) translated 5 units down is:**
\[
f(x) = 3^x - 5
\]
**Answer:** \( f(x)=3^{x}-5 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To translate the function \( f(x) = 3^x \) down by 5 units, you simply subtract 5 from the entire function. Therefore, the correct representation of the translation would be \( f(x) = 3^x - 5 \). This effectively lowers all output values of the function by 5. Understanding translations like this is important when visualizing functions. It not only helps you grasp how the graph shifts but also enhances your ability to manipulate and predict function behaviors in more complex scenarios! So, next time you’re working with functions, remember that adding or subtracting values shifts the graph up or down, while adding or subtracting inside the function affects the left or right movement!
