https://r2.erweima.ai/mathbot/image/1727207848880q90862ou.pdf

It looks like you've uploaded a document with some mathematical problems. Here’s a breakdown of the questions from the document:

1. **Part (a)**: You're asked to make a table of values to graph three functions: 
   - \( y = x \)
   - \( y = -x \)
   - \( y = -|x| \)
   After graphing, you are to observe any patterns and relationships between these graphs.

2. **Part (b)**: The second part involves calculating the area under the curve \( y = x \), bounded by the x-axis and \( x = 4 \), which is given as \( \frac{16}{3} \) square units. Then, using graph transformations, you need to deduce the area under the curve \( y = -x \), bounded below by the x-axis and on the left by \( x = -4 \). You are also required to justify your answer.

Would you like assistance with solving one of these parts, or a detailed explanation of the process?

### Follow-up questions for deeper understanding:
1. How do graph transformations affect the shapes and areas of functions?
2. What method would you use to calculate the area under a curve, and how does it apply here?
3. How are the functions \( y = x \), \( y = -x \), and \( y = -|x| \) related geometrically?
4. How do reflections over the x-axis or y-axis influence the graph's area properties?
5. Why is the area under \( y = -x \) likely to be similar to that of \( y = x \)?

**Tip**: When dealing with graph transformations, always consider whether the transformation preserves, reflects, or dilates the area under the curve.

Make a table of values to graph y = x. Then on the same set of coordinate axes graph y = -x and y = -|x|. What pattern do you observe? How are the graphs related? Calculate the area under y = x and y = -x bounded by the x-axis.

Explore graphing the functions y = x, y = -x, and y = -|x|. Learn how to observe relationships between the graphs and calculate the area under these curves using graph transformations and integration techniques. Suitable for Grades 9-12.

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