https://file.aiquickdraw.com/mathbot/file/17255987607038txj1j3o.jpg

The image contains three function analysis problems, each asking to identify the parent function, describe the transformations, and find the domain, range, intervals of increasing and decreasing, and fixed points. Here's a detailed solution for each:

### Problem 4:
**Function:** \( f(x) = -\frac{1}{2} |x - 5| - 3 \)

1. **Parent Function:** \( |x| \)
2. **Transformations:**
   - Reflect over the x-axis due to the negative sign.
   - Vertical compression by a factor of \(\frac{1}{2}\).
   - Horizontal shift right by 5 units.
   - Vertical shift down by 3 units.
3. **Domain:** All real numbers (\(-\infty, \infty\))
4. **Range:** \( y \leq -3 \)
5. **Intervals:**
   - **Increasing Interval:** None
   - **Decreasing Interval:** \((-\infty, \infty)\)
6. **Fixed Points:** None

### Problem 5:
**Function:** \( f(x) = -\frac{1}{2}(2x - 1) + 1 \)

1. **Parent Function:** Linear \( y = x \)
2. **Transformations:**
   - Reflect over the x-axis.
   - Horizontal stretch/compression and shift depending on the inside term.
   - Shifted vertically up by 1 unit.
3. **Domain:** All real numbers (\(-\infty, \infty\))
4. **Range:** All real numbers (\(-\infty, \infty\))
5. **Intervals:**
   - **Increasing Interval:** None (if reflection occurs)
   - **Decreasing Interval:** All real numbers.
6. **Fixed Points:** Solving for intersections.

### Problem 6:
**Function:** \( f(x) = \) (The function appears unclear; if it's exponential or quadratic, specifics may vary.)

1. **Parent Function:** Depending on type (quadratic \( x^2 \), exponential \( e^x \), etc.)
2. **Transformations:**
   - Vertical shifts or reflections based on coefficients.
   - Stretches or compressions as denoted by scale factors.
3. **Domain:** Typically all real numbers or positive \( x > 0 \).
4. **Range:** Bounded by horizontal shifts or transformations.
5. **Intervals:**
   - Defined by increasing or decreasing depending on the transformation.
6. **Fixed Points:** Determined by specific equal conditions between transformations and original.

Do you need further clarification on each function's graphical analysis?

### Additional Questions:
1. How would you determine the fixed points algebraically for Problem 4?
2. Can you identify the differences in range and behavior between absolute value and quadratic functions?
3. How does a vertical compression affect the intervals of increase or decrease?
4. What are the key differences between vertical and horizontal transformations?
5. Why does reflection across the x-axis change the interval behavior in Problem 5?

**Tip:** When analyzing transformations, breaking down each step (shift, stretch, reflection) systematically helps in constructing accurate graphs and intervals.

The user submitted a query involving function transformations, asking for the identification of parent functions, transformations, and analysis of domain, range, intervals of increase and decrease, and fixed points.

Analyze function transformations including absolute value, linear, and quadratic functions. Explore parent functions, vertical and horizontal shifts, reflections, and compressions with detailed domain, range, and interval behavior. Suitable for students in Grades 9-12.

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