Here are some typical transformation problems on a graph, along with their answers:

1. Vertical Translation

Consider the function $f(x) = x^2$. Apply a vertical translation of 3 units upwards.

Solution: The new function after the transformation is $g(x) = x^2 + 3$. This translates every point of the graph of $f(x)$ upwards by 3 units.

---

2. Horizontal Translation

Consider the function $f(x) = \sqrt{x}$. Apply a horizontal translation 2 units to the right.

Solution: The new function is $g(x) = \sqrt{x - 2}$. This shifts the graph of $f(x)$ 2 units to the right.

---

3. Reflection across the x-axis

Consider the function $f(x) = \sin(x)$. Reflect the graph across the x-axis.

Solution: The new function is $g(x) = -\sin(x)$. This flips the graph of $f(x)$ over the x-axis, making all positive values negative and vice versa.

---

4. Vertical Stretch

Consider the function $f(x) = x^3$. Apply a vertical stretch by a factor of 2.

Solution: The new function is $g(x) = 2x^3$. This makes the graph steeper, as all y-values are doubled.

---

5. Horizontal Compression

Consider the function $f(x) = \cos(x)$. Apply a horizontal compression by a factor of 1/2.

Solution: The new function is $g(x) = \cos(2x)$. The graph is compressed horizontally, meaning the graph cycles through its period twice as fast as before.

---

Would you like more details or have any questions on these? Here are some related questions:

1. How do you determine the direction of a horizontal translation?
2. What happens to the graph of a function when it undergoes a vertical stretch?
3. What is the effect of reflecting a function across the y-axis instead of the x-axis?
4. Can you explain how a horizontal stretch differs from a vertical stretch?
5. How does adding or subtracting a constant inside a function's argument affect its graph?

Tip: Always pay attention to whether the transformation is applied inside or outside the function to determine if it affects the x-values (horizontal changes) or y-values (vertical changes).