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:

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).

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 $-14$.

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 $5$.

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

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!