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Let's go through the problems in this image:

### Problems 1-4: Comparing Graphs of Two Functions
1. **\( f(x) = x \), \( g(x) = -2x + 3 \)**
   - \( g(x) \) is a linear function as well, but it is a reflection of \( f(x) \) across the x-axis (due to the negative sign) and a vertical stretch by a factor of 2. Additionally, it's shifted upward by 3 units.

2. **\( f(x) = |x| \), \( g(x) = -\frac{1}{2}|x| + 3 \)**
   - \( g(x) \) involves a vertical compression of \( f(x) \) by a factor of \( \frac{1}{2} \) and reflection across the x-axis. Then, it's shifted up by 3 units.

3. **\( f(x) = x^2 \), \( g(x) = 3(x + 4)^2 - 1 \)**
   - The graph of \( g(x) \) is the graph of \( f(x) = x^2 \) shifted to the left by 4 units (due to \( +4 \)), vertically stretched by a factor of 3, and shifted downward by 1 unit.

4. **\( f(x) = 3x \), \( g(x) = \frac{3}{4}(x - 2) \)**
   - \( g(x) \) involves a horizontal shift to the right by 2 units and a vertical compression by a factor of \( \frac{3}{4} \).

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### Problems 5-8: Identifying Parent Function, Values of \( a \), \( h \), \( k \) and Graphing
For each of the following functions, we'll identify the parent function and values of \( a \), \( h \), and \( k \), and describe the transformations.

5. **\( y = x^2 - 3 \)**
   - Parent function: \( y = x^2 \)
   - \( a = 1 \) (no vertical stretch/compression)
   - \( h = 0 \) (no horizontal shift)
   - \( k = -3 \) (shifted down by 3 units)

6. **\( y = -3|x + 2| + 4 \)**
   - Parent function: \( y = |x| \)
   - \( a = -3 \) (vertical stretch by 3 and reflection across x-axis)
   - \( h = -2 \) (shifted to the left by 2 units)
   - \( k = 4 \) (shifted up by 4 units)

7. **\( y = \frac{2}{3}(x - 1) + 2 \)**
   - Parent function: \( y = x \) (linear)
   - \( a = \frac{2}{3} \) (vertical compression by \( \frac{2}{3} \))
   - \( h = 1 \) (shifted to the right by 1 unit)
   - \( k = 2 \) (shifted up by 2 units)

8. **\( y = -(3 - x)^2 - 1 \)**
   - Parent function: \( y = x^2 \)
   - \( a = -1 \) (reflection across x-axis, no vertical stretch/compression)
   - \( h = 3 \) (shifted to the right by 3 units)
   - \( k = -1 \) (shifted down by 1 unit)

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If you need detailed graphing or further explanation, feel free to ask!

### Here are some related questions to explore:
1. How do vertical shifts affect the graph of a function?
2. What is the effect of multiplying a function by a negative number?
3. How can horizontal shifts be identified in function transformations?
4. How do you determine whether a function is stretched or compressed?
5. What is the geometric meaning of reflecting a function across the x-axis?

#### Tip:
Always pay close attention to the signs of \( h \) and \( k \) when determining shifts in a function's graph.

Comparing graphs and identifying parent functions for different equations and transformations

This problem set involves comparing the graphs of different functions and identifying transformations such as shifts, reflections, stretches, and compressions. Learn how to analyze the relationship between parent functions and transformations for linear, quadratic, and absolute value functions. Suitable for high school students in Grades 9-12.

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