Question

Describe all of the transformations of f(x)=14⋅3−x+5+8 . (ie. left/right, up/down. stretch/compressed, reflected?)
transformation options line 1: left, right, neither line 2: up, down, neither line 3: stretch, compress, neither line
4: reflected across x-axis
reflected across y-axis
neither
both

By how much?
line 1: 1/4, 3, 5, 8
line2:  1/4, 3, 5, 8
line 3:  1/4, 3, 5, 8

Answer 1

To analyze the function \( f(x) = 14 \cdot 3^{-x} + 5 + 8 \), we can first simplify it:

\[
f(x) = 14 \cdot 3^{-x} + 13
\]

Now, let's break down the transformations step by step:

1. **Reflections**:
- The term \( 3^{-x} \) indicates a reflection across the y-axis because we can rewrite \( 3^{-x} \) as \( \frac{1}{3^x} \), which reflects the function.
- So, it is reflected across the y-axis.

2. **Vertical Stretch/Compression**:
- The coefficient \( 14 \) in front of \( 3^{-x} \) indicates a vertical stretch by a factor of \( 14 \).

3. **Shifts**:
- The constants \( +5 + 8 \) sum to \( +13 \), indicating a vertical shift upward.
- There is no horizontal shift indicated by the function as the transformation \( 3^{-x} \) does not contain any \( +c \) or \( -c \) affecting \( x \).

4. **Horizontal Shift**:
- There is no left or right shift in this function.

Putting this information in the requested format:

**Transformations**:
- Line 1 (Left/Right): **Neither**
- Line 2 (Up/Down): **Up** (by 13 units)
- Line 3 (Stretch/Compress): **Stretch** (by a factor of 14)
- Reflected across the x-axis: **Neither**
- Reflected across the y-axis: **Reflected**

**By how much?**:
- Line 1: **Neither**
- Line 2: **13**
- Line 3: **14**

So, summarizing the transformations:

- Transformations:
- Left/Right: Neither
- Up/Down: Up by 13
- Stretch/Compress: Stretch by a factor of 14
- Reflected across x-axis: Neither
- Reflected across y-axis: Reflected

I hope this breakdown helps clarify the transformations of the function!