https://file.aiquickdraw.com/mathbot/file/17270389487583u897aob.jpg

In this image, we are asked to find the formula for \( g(x) \) in terms of \( f(x) \), where \( f \) is the blue graph on the left, and \( g \) is the red graph on the right. 

### Step-by-step Analysis:
1. **Look at the shape of the graphs**: Both graphs represent piecewise linear functions. \( f(x) \) has a sharp "V" shape centered at \( (0, 0) \), and \( g(x) \) appears to be a vertically shifted and slightly transformed version of \( f(x) \).

2. **Check for shifts or transformations**: 
   - The graph of \( f(x) \) has vertices at \( (-2, -4) \), \( (0, 0) \), and \( (2, -4) \). It is symmetric about the y-axis.
   - The graph of \( g(x) \) has the same vertex pattern, but the function appears to be shifted down by 2 units compared to \( f(x) \).

3. **Conclude the transformation**:
   From the observation that \( g(x) \) looks exactly like \( f(x) \) but shifted downward by 2 units, we can hypothesize:
   \[
   g(x) = f(x) - 2
   \]
   This transformation corresponds to a vertical shift of the graph of \( f(x) \) down by 2 units.

### Final Answer:
\[
g(x) = f(x) - 2
\]

This is the formula for \( g \) in terms of \( f \).

Would you like further explanation, or do you have any specific questions?

Here are 5 related questions to expand your understanding:
1. How can you identify horizontal transformations from graphs?
2. What are other types of transformations, such as stretching or reflection?
3. How does shifting a function horizontally differ from vertically?
4. Can you express \( f(x) \) in terms of a piecewise function based on its graph?
5. How would the formula for \( g(x) \) change if the graph of \( g(x) \) was a reflection of \( f(x) \)?

**Tip**: Always check for both vertical and horizontal shifts when comparing two similar graphs.

Discover how to derive the formula for the graph g(x) based on transformations of the graph f(x). This guide explains how a vertical shift can modify the function's output, illustrated through piecewise linear functions. Suitable for high school students learning about graph transformations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation