Describe all of the transformations of f(x)=14⋅3−x+5+8 . (ie. left/right, up/down. stretch/compressed, reflected?)

Question

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!