Here are the graphs for each of the functions from the exercises:

1. **Linear Equations:**
   - \( y = \frac{1}{2}x + 2 \) (Top left, in blue)
   - \( y = x \) (Top right, in orange)

2. **Quadratic Equation:**
   - \( y = \frac{1}{4}x^2 \) (Middle left, in green)

3. **Absolute Value Function:**
   - \( y = |x| - 1 \) (Middle right, in red)

4. **Square Root Functions:**
   - \( y = \sqrt{x} - 6 \) (Bottom left, in purple)
   - \( y = \sqrt{x} + 2 \) (Bottom middle, in brown)

5. **Rational Functions:**
   - \( y = \frac{1}{x} \) (Bottom left, in cyan)
   - \( y = \frac{1}{x^2 + 2} \) (Bottom right, in magenta)

These plots should help visualize how each function behaves over a range of \(x\) values.

Do you have any questions about these graphs or how they were plotted?

Here are 5 follow-up questions for further exploration:
1. How would the graph of \( y = \frac{1}{2}x + 2 \) change if the slope or intercept were different?
2. What is the effect of changing the coefficient of \(x^2\) in the quadratic equation on the width of the parabola?
3. How does shifting the square root function up or down affect its graph?
4. What are the asymptotes of the rational function \( y = \frac{1}{x} \)?
5. What happens to the graph of \( y = |x| - 1 \) if we change the subtraction to addition?

**Tip:** For rational functions like \( y = \frac{1}{x} \), always watch out for vertical asymptotes where the function is undefined (like \( x = 0 \)).

Explore graphical representations of various mathematical functions, including linear equations, quadratic equations, absolute value functions, square root functions, and rational functions. Each graph is analyzed to understand the behavior of these functions over different ranges of x values. Ideal for high school students studying algebra and functions.

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