Question 1: Write the rule for the transformation that shifts the cubic graph of y = x^3 3 units right and 2 units down.

Answer: y = (x - 3)^3 - 2

Question 2: Plot the graph of y = (x + 2)^3 on the grid below. Describe the transformation from y = x^3.

Answer: Translation 2 units left

Question 3: Reflect the cubic graph y = x^3 across the y-axis. Write the new equation.

Answer: y = -x^3

Question 4: Identify the transformation: y = 2(x - 1)^3 + 4.

Answer: Vertical stretch by a factor of 2, shift right 1, shift up 4

Question 5: Determine the effect of the transformation y = (x/2)^3 on the original y = x^3 graph.

Answer: Horizontal stretch by a factor of 2

Question 6: A cubic graph is reflected across the x-axis and shifted 3 units left. Write the equation of the transformed graph.

Answer: y = - (x + 3)^3

Question 7: A graph of y = x^3 is stretched vertically by a factor of 3 and then shifted down 5 units. Write the new equation.

Answer: y = 3x^3 - 5

Question 8: A student incorrectly states that shifting y = x^3 right 2 units and up 3 units results in y = (x - 2)^3 + 3. Identify and correct the mistake.

Answer: The correct transformation is y = (x - 2)^3 + 3; no mistake.

Question 9: Construct the cubic graph by applying a reflection across the y-axis, a vertical stretch by 2, and a shift 4 units right. Write the equation of the resulting graph.

Answer: y = -2(x - 4)^3

Question 10: Identify the type of transformation and the order in which it is applied if a graph of y = x^3 is first shifted left 3 units, then reflected across the x-axis, and finally vertically stretched by 3.

Answer: Order: shift left 3, reflect across x-axis, stretch vertically by 3

Question 11: A common mistake is to forget the order of transformations. Explain why changing the order of shifting and reflecting might lead to a different graph.

Answer: Because transformations like shifting and reflecting are not commutative; changing their order changes the final position and orientation of the graph.