https://file.aiquickdraw.com/mathbot/file/17290602810652aida1st.jpg

To solve this problem, we are tasked with using the graph of \( y = f(x) \) to graph the function \( g(x) = -f(x) - 1 \).

### Step-by-Step Explanation:

1. **Understanding the given graph:**
   - The graph of \( y = f(x) \) is a horizontal line segment from \( x = -3 \) to \( x = 3 \), lying on the line \( y = -3 \).

2. **Transformation for \( g(x) = -f(x) - 1 \):**
   - The function \( g(x) = -f(x) - 1 \) is obtained from \( f(x) \) by applying two transformations:
     - **Reflection across the x-axis:** This changes the sign of \( f(x) \), so the values of \( y \) will become positive where they were negative, and vice versa. For the given graph, since \( f(x) = -3 \), \( -f(x) = 3 \) for the same \( x \)-values.
     - **Vertical translation by -1:** After reflecting, we shift the graph down by 1 unit. This changes the value of \( y = 3 \) to \( y = 2 \) for the entire segment.

3. **Applying the transformation:**
   - The original line segment from \( x = -3 \) to \( x = 3 \), which was at \( y = -3 \), will first reflect to \( y = 3 \), and then shift down by 1 unit to \( y = 2 \).

### Matching the correct graph:
- Looking at the options, the correct graph will have a horizontal line segment from \( x = -3 \) to \( x = 3 \), positioned at \( y = 2 \).

- **Correct answer: Option D** is the graph that corresponds to this transformation.

Would you like more details, or have any questions?

Here are 5 related questions:
1. What other transformations can we apply to \( f(x) \) besides reflection and translation?
2. How does vertical stretching or shrinking affect the graph of a function?
3. Can you explain horizontal translations and how they would modify \( f(x) \)?
4. What happens to the graph if we change the function to \( g(x) = f(x) + 2 \)?
5. How would the graph look if we applied the transformation \( g(x) = -f(x) + 1 \)?

**Tip:** When reflecting across the x-axis, just change the sign of all the \( y \)-coordinates without altering the \( x \)-coordinates.

Use the graph of y = f(x) to graph the function g(x) = -f(x) - 1.

Learn how to graph the function g(x) = -f(x) - 1 using transformations. This problem explores reflection across the x-axis and vertical translation. A step-by-step solution is provided 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