Question

If f(x) =7^x
is the parent function, and f(x) = -7^1/5 x +3

, describe the transformations that took place.

A
vertical shift 3 units up

B
reflected across the y-axis

C
reflected across the x-axis

D
vertical shift 3 units down

E
vertical stretch of 5

F
horizontal stretch of 5

Answer 1

To analyze the transformation from the parent function \( f(x) = 7^x \) to the new function \( f(x) = -7^{\frac{1}{5}x} + 3 \), we can break it down step by step.

1. **Starting from the parent function**: \( f(x) = 7^x \)

2. **Horizontal Compression**: The term \( 7^{\frac{1}{5}x} \) indicates a horizontal stretch. Since the exponent is multiplied by \( \frac{1}{5} \) (which is less than 1), this compresses the graph horizontally.

3. **Reflection Across the x-axis**: The negative sign in front of \( 7^{\frac{1}{5}x} \) indicates that the graph is reflected across the x-axis.

4. **Vertical Shift**: The term \( +3 \) indicates that the entire graph is shifted up 3 units.

Putting that together, the transformations are:

- A horizontal stretch of 5 (which corresponds to the \( \frac{1}{5} \) in the exponent),
- A reflection across the x-axis (due to the negative sign),
- A vertical shift 3 units up (due to the \( +3 \)).

From the options provided:

- A: Vertical shift 3 units up → **True**
- B: Reflected across the y-axis → **False**
- C: Reflected across the x-axis → **True**
- D: Vertical shift 3 units down → **False**
- E: Vertical stretch of 5 → **False**
- F: Horizontal stretch of 5 → **False**

Thus, the correct transformations are:

- Vertical shift 3 units up (A)
- Reflected across the x-axis (C)