Question 1: Write the equation of the cubic graph that results from shifting y=x^3 one unit right and two units down.

Answer: y=(x-1)^3 - 2

Question 2: Construct the graph of y= (x+2)^3 and describe the transformation applied to y=x^3.

Answer: Shifted 2 units left.

Question 3: If y=x^3 is reflected across the x-axis, what is the new equation?

Answer: -x^3

Question 4: Plot the graph of y=2(x-1)^3 + 3 on the grid provided.

Answer: See full answer key

Question 1: A cubic graph is reflected across the y-axis and then shifted 3 units up. Write the resulting equation if the original is y=x^3.

Answer: -x^3 + 3

Question 2: Describe the sequence of transformations needed to convert y=x^3 into y=0.5(x+2)^3 - 4.

Answer: Horizontal shift 2 units left, vertical compression by factor 0.5, shift 4 units down.

Question 3: Determine the effect of multiplying y=x^3 by -2 on the graph's shape and position.

Answer: Reflected across x-axis and vertically stretched by factor 2.

Question 4: Construct the graph of y= (x-3)^3/2 and explain the transformations involved.

Answer: Horizontal shift 3 units right; vertical compression by factor 1/2.

Question 1: A cubic function models the volume of a container as a function of height. If the basic volume function is y=x^3, and the container is designed to be wider at the top, described by y=(x-2)^3 + 5, interpret the transformation and its possible real-world meaning.

Answer: Horizontal shift 2 units right (wider at top), upward shift 5 units.

Question 1: Given y=x^3, find the equation of the graph obtained by reflecting across the x-axis, then horizontally stretching by a factor of 3, and finally shifting 4 units down.

Answer: -(3x)^3 - 4

Question 2: Prove that the composition of two transformations, a reflection across the y-axis and a vertical stretch by factor 2, results in the function y= -2x^3.

Answer: Reflection: y= -x^3; stretch: y= -2x^3.

Question 1: Identify the transformation: y=x^3 reflected across y = x and then shifted 1 unit left.

Answer: Reflected across y=x and shifted left 1.

Question 2: Construct the graph of y= -0.5(x-4)^3 + 2 and describe the combined transformations.

Answer: Reflection across x-axis, vertical compression by 0.5, shift 4 units right, shift 2 units up.

Question 1: A student claims that shifting y=x^3 two units right results in y=(x+2)^3. Is this correct? Explain and correct if necessary.

Answer: Incorrect; shift right 2 units gives y=(x-2)^3.