The problem on the page involves applying transformations to given linear functions and graphing them. Let's go over each subpart of the question:

Part A

• Original Function: $f(x) = 3x + 3$
• Transformation: Graph $f(x) - 3$.

To apply this transformation, we subtract 3 from the $y$-values of the original function. This is a vertical shift downward by 3 units.

Part B

• Original Function: $f(x) = x + 7$
• Transformation: Graph $-2f(x)$.

This transformation involves two steps:

1. Multiply the original function by -2, which reflects the graph across the x-axis and vertically stretches it by a factor of 2. $f(x) = x + 7 \quad \Rightarrow \quad -2f(x) = -2(x + 7) = -2x - 14$

Part C

• Original Function: $f(x) = 2x + 1$
• Transformation: Graph $2f(x) + 3$.

This transformation scales the graph vertically by a factor of 2 and then shifts it upwards by 3 units. The equation becomes: $f(x) = 2x + 1 \quad \Rightarrow \quad 2f(x) + 3 = 2(2x + 1) + 3 = 4x + 2 + 3 = 4x + 5$

For each case, the transformed graphs need to be sketched accordingly.

Would you like further clarification on any of the transformations or specific details for graphing them? Here are 5 questions you can think about:

1. What is the effect of multiplying a function by a negative number on its graph?
2. How do vertical shifts and stretches affect the slope of a line?
3. What does reflecting a graph across the x-axis do to its points?
4. Can you explain the difference between a vertical and a horizontal transformation?
5. How does adding a constant to a function's output affect the function's graph?

Tip: When dealing with function transformations, always consider how each operation (e.g., addition, multiplication) modifies the graph, either by shifting, stretching, or reflecting.