https://file.aiquickdraw.com/mathbot/file/1726680785467mkixd2lo.jpg

The problem presents two functions, \( f(x) = |x| \) and \( g(x) = |x| - 5 \), and asks to graph them together and describe how the graph of \( g(x) \) is related to that of \( f(x) \).

1. **Function \( f(x) = |x| \)** is the basic absolute value function. Its graph is a V-shaped curve with the vertex at \( (0, 0) \) and slopes of 1 and -1 for positive and negative \( x \)-values, respectively.

2. **Function \( g(x) = |x| - 5 \)** is a vertical translation of the graph of \( f(x) \). The effect of subtracting 5 from \( f(x) \) is that the entire graph of \( f(x) \) is shifted downward by 5 units.

Thus, the graph of \( g(x) \) will have the same V-shape but will have its vertex at \( (0, -5) \) instead of \( (0, 0) \).

### Graphing and Relation:
- The graph of \( g(x) \) is the same as \( f(x) \), but every point on the graph of \( f(x) \) is shifted 5 units down to get the graph of \( g(x) \).
  
Would you like to explore the graphs more or have any questions? Here are some related questions for deeper understanding:

1. How would the graph change if the transformation were \( g(x) = |x| + 5 \)?
2. What is the geometric significance of adding/subtracting constants outside the absolute value function?
3. What would happen if we transformed \( f(x) = |x| \) horizontally, such as \( f(x) = |x - 3| \)?
4. How do reflections affect the graph of \( f(x) = |x| \), such as \( f(x) = -|x| \)?
5. Can we combine horizontal and vertical transformations in one function? How would it look?

**Tip:** Vertical translations modify the \( y \)-coordinate of all points on the graph by the same amount.

Graph the given functions, f(x) = |x| and g(x) = |x| - 5, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of f.

Learn how to graph absolute value functions f(x) = |x| and g(x) = |x| - 5, focusing on vertical translations. This problem explores key graph transformation concepts suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation