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Let's analyze the given question and solve it step by step.

The problem involves transforming the given functions and graphing the results. Each function has a specified transformation. Let's solve each part:

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### **Part (a):**
**Given Function:** \( f(x) = 3x + 3 \)  
**Transformation:** \( f(x) - 3 \)

1. **Transformation Explanation:** Subtracting 3 from \( f(x) \) shifts the graph **down** by 3 units.  
   The new function is:
   \[
   f(x) - 3 = 3x + 3 - 3 = 3x
   \]
2. **Graph:** The slope remains \( 3 \), but the y-intercept changes to \( 0 \). The graph now passes through the origin (0, 0).

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### **Part (b):**
**Given Function:** \( f(x) = x + 7 \)  
**Transformation:** \( -2f(x) \)

1. **Transformation Explanation:** First, multiply \( f(x) \) by -2. This reflects the graph across the x-axis (due to the negative) and stretches it vertically by a factor of 2.  
   The new function is:
   \[
   -2f(x) = -2(x + 7) = -2x - 14
   \]
2. **Graph:** The slope is now \( -2 \), and the y-intercept is \( -14 \).

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### **Part (c):**
**Given Function:** \( f(x) = 2x + 1 \)  
**Transformation:** \( 2f(x) + 3 \)

1. **Transformation Explanation:** Multiply \( f(x) \) by 2, and then add 3.  
   The new function is:
   \[
   2f(x) + 3 = 2(2x + 1) + 3 = 4x + 2 + 3 = 4x + 5
   \]
2. **Graph:** The slope is \( 4 \), and the y-intercept is \( 5 \).

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Would you like me to sketch the transformations or explain any step in more detail?

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### Related Questions:
1. What happens to a function if you add a constant inside versus outside the parentheses?  
2. How does multiplying by a negative number affect the graph of a function?  
3. Why does changing the slope stretch or compress the graph?  
4. Can you explain why subtracting a constant shifts the graph vertically?  
5. How would the transformations change if they were applied in reverse order?

### Tip:  
Always rewrite the transformed function algebraically first—it helps you spot changes in slope, intercept, and transformations!

For each of the following, a function has been given along with its graph. Perform the specified transformation of the function and graph.

Solve and graph linear function transformations, such as f(x) = 3x + 3 shifted downward or scaled. Understand concepts like vertical shifts, reflections, and slope scaling. Suitable for Grades 9-11.

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