Let's solve this step by step.

1. Fill in the blanks for the following statements:

a) Given a function $f(x)$:

• A new function $g(x) = Af(x)$, where $A$ is a constant, is a vertical stretch or compression of the function $f(x)$.

• If $A > 1$, then the graph will be vertically stretched by a factor of $A$.

• If $0 < A < 1$, then the graph will be vertically compressed by a factor of $A$.

• If $A < 0$, then there will be a reflection of a vertical stretch or compression, and either a vertical stretch or compression.

b) Given a function $f(x)$:

• A new function $g(x) = f(Bx)$, where $B$ is a constant, is a horizontal stretch or compression of the function $f(x)$.

• If $0 < B < 1$, then the graph will be horizontally stretched by a factor of $\frac{1}{B}$.

• If $B > 1$, then the graph will be horizontally compressed by a factor of $\frac{1}{B}$.

• If $B < 0$, then there will be a reflection of a horizontal stretch or compression, and either a horizontal stretch or compression.

2. Graph transformations:

You are asked to sketch and describe the transformations for two functions $g(x)$ and $h(x)$ based on the graph of $f(x)$.

• For $g(x) = \frac{1}{2} f(x)$:

  • This represents a vertical compression by a factor of 2 (since $\frac{1}{2}$ is less than 1).

  • Transformation:

    • The graph of $f(x)$ will be vertically compressed, meaning the y-values of the original graph will be multiplied by $\frac{1}{2}$.
    • Sketch: The new graph will look like the original graph, but "flatter," with all points scaled down to half their original y-value.

• For $h(x) = f\left(\frac{1}{2} x\right)$:

  • This represents a horizontal stretch by a factor of 2 (since the coefficient $\frac{1}{2}$ inside the function causes a horizontal stretch).

  • Transformation:

    • The graph of $f(x)$ will be horizontally stretched, meaning the x-values of the original graph will be multiplied by 2.
    • Sketch: The new graph will look like the original, but stretched horizontally, making it wider.

Do you have any questions or want more details?

Here are 5 related questions to further explore this topic:

1. How do vertical shifts affect the graph of $f(x)$?
2. What happens if $A$ or $B$ is negative in the transformations?
3. How can you determine if a graph has undergone a reflection?
4. What is the difference between vertical and horizontal transformations in terms of their impact on the graph's shape?
5. How would you describe the transformation $g(x) = 2f(3x)$?

Tip: When working with transformations, always look at how constants inside or outside the function affect the scaling, shifting, or reflection of the graph.