Answered

Instructions: Determine the horizontal stretch, compression, or reflection of the functions as practiced. In the next set of problems, all types of transformations will be combined. Remember, it may help to imagine the stretching or compressing of the functions horizontally.

1. What is the effect on the graph of the function [tex]f(x) = 3^x[/tex] when [tex]f(x)[/tex] is replaced with [tex]f\left(\frac{1}{5} x\right)[/tex]?
- This is a \_\_\_\_\_\_
- Horizontal reflection over the y-axis? \_\_\_\_\_\_

2. How is the graph of the function [tex]y = 2^{6x}[/tex] transformed from the function [tex]y = 2^x[/tex]?
- This is a \_\_\_\_\_\_
- Horizontal reflection over the y-axis? \_\_\_\_\_\_

3. How is the graph of the function [tex]y = 2^{-3x}[/tex] transformed from the function [tex]y = 2^x[/tex]?
- This is a \_\_\_\_\_\_
- Horizontal reflection over the y-axis? \_\_\_\_\_\_



Answer :

GinnyAnswer
Let's go through each question step by step.

### 1. Transformation of [tex]\( f(x) = 3^x \)[/tex]

- Original Function: [tex]\( f(x) = 3^x \)[/tex]
- Modified Function: [tex]\( f\left(\frac{1}{5} x\right) \)[/tex]

When [tex]\( f(x) \)[/tex] is replaced with [tex]\( f\left(\frac{1}{5} x\right) \)[/tex]:

- This is a horizontal stretch by a factor of 5.

No horizontal reflection is involved here.

### 2. Transformation of [tex]\( y = 2^{6x} \)[/tex]

- Original Function: [tex]\( y = 2^x \)[/tex]
- Modified Function: [tex]\( y = 2^{6x} \)[/tex]

When [tex]\( y = 2^x \)[/tex] is replaced with [tex]\( y = 2^{6x} \)[/tex]:

- This is a horizontal compression by a factor of [tex]\(\frac{1}{6}\)[/tex].

No horizontal reflection is involved here.

### 3. Transformation of [tex]\( y = 2^{-3x} \)[/tex]

- Original Function: [tex]\( y = 2^x \)[/tex]
- Modified Function: [tex]\( y = 2^{-3x} \)[/tex]

When [tex]\( y = 2^x \)[/tex] is replaced with [tex]\( y = 2^{-3x} \)[/tex]:

- This is a horizontal compression by a factor of [tex]\(\frac{1}{3}\)[/tex].
- Additionally, this involves a horizontal reflection over the y-axis.

### Summary of the Effects:

- For the function [tex]\( f(x) = 3^x \)[/tex] replaced with [tex]\( f\left(\frac{1}{5} x\right) \)[/tex]
- This is a [tex]\( \boxed{\text{horizontal stretch}} \)[/tex]
- Horizontal reflection over [tex]\( y \)[/tex]? [tex]\( \boxed{\text{False}} \)[/tex]

- For the function [tex]\( y = 2^{6x} \)[/tex] transformed from [tex]\( y = 2^x \)[/tex]
- This is a [tex]\( \boxed{\text{horizontal compression}} \)[/tex]
- Horizontal reflection over [tex]\( y \)[/tex]? [tex]\( \boxed{\text{False}} \)[/tex]

- For the function [tex]\( y = 2^{-3x} \)[/tex] transformed from [tex]\( y = 2^x \)[/tex]
- This is a [tex]\( \boxed{\text{horizontal compression}} \)[/tex]
- Horizontal reflection over [tex]\( y \)[/tex]? [tex]\( \boxed{\text{True}} \)[/tex]

Other Questions