let f(x) = |x|, Vertical compression  by a factor of 2/3 followed by a horizontal translation 4 units to the left and a reflection across the x axis



To find the transformed function based on the transformations applied to \( f(x) = |x| \), we need to apply each transformation step by step:

### 1. **Vertical Compression by a Factor of \( \frac{2}{3} \)**:
- A vertical compression by a factor of \( \frac{2}{3} \) modifies the function by multiplying the entire function by \( \frac{2}{3} \). 
- So, the function becomes:
  \[
  f(x) = \frac{2}{3}|x|
  \]

### 2. **Horizontal Translation 4 Units to the Left**:
- A horizontal translation 4 units to the left shifts the input \( x \) by adding 4 inside the function.
- The function now becomes:
  \[
  f(x) = \frac{2}{3}|x + 4|
  \]

### 3. **Reflection Across the \( x \)-Axis**:
- A reflection across the \( x \)-axis changes the sign of the entire function.
- The final transformed function is:
  \[
  f(x) = -\frac{2}{3}|x + 4|
  \]

### Final Answer:
The transformed function is:
\[
f(x) = -\frac{2}{3}|x + 4|
\]

Would you like further details or have any questions about this transformation process?

---

Here are 5 related questions to consider:
1. What happens to the graph of \( f(x) = |x| \) when it's vertically stretched instead of compressed?
2. How would you express a horizontal translation 4 units to the right instead of to the left?
3. How does a reflection across the y-axis change the equation of \( f(x) = |x| \)?
4. Can you describe the effect of combining both vertical and horizontal transformations at the same time?
5. How would the function change if we applied a horizontal compression instead of a vertical one?

**Tip:** Always apply horizontal transformations inside the function argument and vertical transformations outside the function to ensure the correct sequence of changes.

Let f(x) = |x|, apply a vertical compression by a factor of 2/3 followed by a horizontal translation 4 units to the left and a reflection across the x-axis.

Learn how to transform the absolute value function f(x) = |x| by applying a vertical compression, horizontal translation, and reflection across the x-axis. This detailed guide walks through each step of the process, providing insights into key mathematical concepts like function transformations and reflections. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation