https://file.aiquickdraw.com/mathbot/file/1727311106677mi4w9syu.jpg

Write the transformation of f(x) to g(x) then write the equation g(x) in terms of f(x)=x

To understand the transformation from \( f(x) \) to \( g(x) \), let's first analyze the graph:

1. **Function \( f(x) \)**: The function \( f(x) \) is a piecewise linear function, forming a "V" shape. This indicates that \( f(x) \) is most likely the absolute value function:
   \[
   f(x) = |x|
   \]
   This fits the graph of \( f(x) \), where the point of the "V" occurs at \( x = 0 \), and the slopes of the two lines forming the "V" are \( 1 \) and \( -1 \).

2. **Function \( g(x) \)**: The function \( g(x) \) appears to be a transformation of \( f(x) \), as it looks like it involves a shift and a reflection.

### Step 1: Reflection
- \( g(x) \) seems to be reflected over the x-axis relative to \( f(x) \). This is a vertical reflection, which changes \( f(x) = |x| \) to \( -f(x) \), meaning:
  \[
  g(x) = -|x|
  \]

### Step 2: Shift
- \( g(x) \) is shifted horizontally to the right by 4 units. To account for this, we replace \( x \) with \( (x - 4) \) in the function:
  \[
  g(x) = -|x - 4|
  \]

### Step 3: Vertical Shift
- Finally, \( g(x) \) is shifted vertically down by 3 units. To incorporate this, we subtract 3 from the function:
  \[
  g(x) = -|x - 4| - 3
  \]

### Conclusion: 
The transformation from \( f(x) = |x| \) to \( g(x) \) involves:
1. A vertical reflection,
2. A horizontal shift to the right by 4 units,
3. A vertical shift down by 3 units.

The equation for \( g(x) \) in terms of \( f(x) = x \) (absolute value function) is:
\[
g(x) = -|x - 4| - 3
\]

Would you like more details on transformations or have any further questions?

Here are five related questions for practice:
1. What happens to a graph when you apply a vertical stretch or compression?
2. How would you write the equation of \( g(x) \) if it were shifted left instead of right?
3. What is the vertex of the function \( g(x) \)?
4. How does reflecting a graph over the y-axis change the equation?
5. What other types of transformations can be applied to a function?

**Tip**: Always track the effect of each transformation (shift, reflection, stretch/compression) one step at a time to avoid confusion.

Write the transformation of f(x) to g(x), then write the equation g(x) in terms of f(x)=|x|.

Learn how to transform the function f(x) = |x| by reflecting it over the x-axis, shifting it 4 units to the right, and 3 units downward to get g(x) = -|x - 4| - 3. This problem involves core concepts of function transformations, including reflection and shifts, suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation