describe the transformation for the following 2 equations: 1-f(-x) and 5-1/6f(x) 

Let's analyze each transformation one by one, starting with \( 1 - f(-x) \).

### 1. \( 1 - f(-x) \):
This transformation involves two key components:
- **Horizontal Reflection**: The term \( f(-x) \) indicates a reflection of the function \( f(x) \) across the y-axis (a horizontal reflection).
- **Vertical Shift**: The term \( 1 - f(-x) \) means the entire function is shifted upwards by 1 unit (since we subtract \( f(-x) \) from 1).

So, the transformation consists of:
1. Reflecting the graph of \( f(x) \) across the y-axis.
2. Shifting the resulting graph upward by 1 unit.

### 2. \( 5 - \frac{1}{6}f(x) \):
This transformation involves a vertical stretch and a vertical shift:
- **Vertical Compression**: The term \( \frac{1}{6}f(x) \) compresses the graph of \( f(x) \) vertically by a factor of \( \frac{1}{6} \) (making it 6 times shorter in height).
- **Vertical Shift**: The term \( 5 - \frac{1}{6}f(x) \) indicates a vertical shift of the function upward by 5 units.

So, the transformation consists of:
1. Compressing the graph of \( f(x) \) vertically by a factor of \( \frac{1}{6} \).
2. Shifting the resulting graph upward by 5 units.

### Summary:
- \( 1 - f(-x) \): Reflect the graph across the y-axis, then shift up by 1.
- \( 5 - \frac{1}{6}f(x) \): Compress the graph vertically by \( \frac{1}{6} \), then shift up by 5.

Would you like further details on any part of this explanation or examples with graphs?

Here are 5 related questions to explore:
1. What happens to the domain and range of \( f(x) \) under these transformations?
2. How would the transformation \( -f(x) \) affect the graph of \( f(x) \)?
3. Can horizontal shifts be combined with these transformations? If so, how?
4. How does a vertical stretch by a factor of 2 differ from a compression by \( \frac{1}{6} \)?
5. What are the combined effects of reflections and shifts on the symmetry of \( f(x) \)?

**Tip**: Always apply horizontal transformations (like reflections or shifts) before vertical ones when graphing functions.

describe the transformation for the following 2 equations: 1 - f(-x) and 5 - 1/6f(x)

Learn how to describe the transformations of functions, including horizontal reflection, vertical compression, and vertical shifts using the equations 1 - f(-x) and 5 - 1/6f(x). This step-by-step explanation covers key concepts like reflecting functions across the y-axis, compressing functions vertically, and shifting functions upwards.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation