Question 1: Describe how the graph of y = x^3 transforms when shifted 3 units to the right and 2 units upwards.

Answer: Shift right 3 units and up 2 units

Question 2: Plot the graph of y = (x - 2)^3 + 4 on the grid. Describe the transformation from y = x^3.

Answer: Shift left 2 units and up 4 units

Question 3: If the cubic graph y = x^3 is reflected across the x-axis and then shifted down 5 units, what is its equation?

Answer: -y = x^3 - 5 or y = -x^3 - 5

Question 4: Calculate the transformation parameters for the graph of y = 2(x + 1)^3 - 3, relative to y = x^3.

Answer: Horizontal shift left 1, vertical stretch by factor 2, shift down 3

Question 5: Construct a cubic graph that is reflected in both axes and shifted 4 units right. Write its equation.

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

Question 6: A cubic graph y = x^3 is stretched vertically by a factor of 3 and shifted 2 units up. Write the new equation.

Answer: y = 3x^3 + 2

Question 7: Explain how the graph of y = x^3 changes when reflected across the y-axis and translated 1 unit down.

Answer: Reflect across y-axis: y = -x^3; shift down 1: y = -x^3 - 1

Question 8: A student claims that shifting y = x^3 by adding 5 to x results in y = (x + 5)^3. Is this correct? Explain.

Answer: No, shifting horizontally by 5 units results in y = (x - 5)^3; adding inside the function shifts left, not right.

Question 9: Identify the error in the graph transformation: y = (x - 2)^3 shifted 3 units left and 4 units up, but the graph shows a shift right 2 units and down 4 units.

Answer: The original shift was incorrect; (x - 2)^3 shifts right 2, not left. To shift left 2, it should be y = (x + 2)^3.

Question 10: Challenge: Derive the equation for a cubic graph that is reflected over the x-axis, shifted 5 units right, and vertically stretched by a factor of 4.

Answer: y = -4(x - 5)^3