https://file.aiquickdraw.com/mathbot/file/1726265248374jk000hvc.jpg

The given function is:

\[
m(x) = \frac{1}{3} |x - 3| - 2
\]

This is an absolute value function, and we can use transformations to graph it. Let's break down the transformation process:

1. **Starting with the base function:** The base function is \( |x| \), which forms a "V" shape with a vertex at the origin (0,0).

2. **Horizontal shift:** The expression \( |x - 3| \) indicates a horizontal shift to the right by 3 units. This moves the vertex from (0,0) to (3,0).

3. **Vertical stretch/compression:** The coefficient \( \frac{1}{3} \) in front of the absolute value compresses the graph vertically by a factor of \( \frac{1}{3} \), making the arms of the "V" less steep compared to the base function \( |x| \).

4. **Vertical shift:** The "-2" at the end of the function moves the entire graph down by 2 units. This means the new vertex is at \( (3, -2) \).

5. **Final graph:** The function results in a "V" shape with a vertex at (3, -2), opening upwards, with the slopes of the arms being \( \frac{1}{3} \) and \( -\frac{1}{3} \).

### Comparing to the provided graph:
- The graph you provided has a "V" shape.
- The vertex appears to be correctly located at \( (3, -2) \), matching the transformations.
- The slopes of the lines appear to match the \( \frac{1}{3} \) slope, indicating the compression is correct.

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

Here are 5 related questions for practice:

1. How would the graph change if the vertical stretch factor was \( \frac{1}{2} \) instead of \( \frac{1}{3} \)?
2. What would happen if the expression was \( |x + 3| \) instead of \( |x - 3| \)?
3. How would the graph shift if the equation had \( +2 \) instead of \( -2 \)?
4. What would the graph look like if the absolute value was replaced by a square root \( \sqrt{x-3} \)?
5. How can you find the axis of symmetry of an absolute value function?

**Tip:** Absolute value functions are symmetric around the vertex, making them an excellent tool to explore reflections and transformations in graphs.

Use transformations to graph the function: m(x) = (1/3)|x - 3| - 2

Learn how to graph the function m(x) = (1/3)|x - 3| - 2 using transformations. This problem involves horizontal shifts, vertical compression, and vertical shifts, making it ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation