Lesson Vocabulary

• power function
• constant of proportionality

Problem 1 Transforming y = x 3 y equals , x cubed

What is an equation of the graph of y = x 3 y equals , x cubed  under a vertical compression by the factor 1 2 1 half  followed by a reflection across the x-axis, a horizontal translation 3 units to the right, and then a vertical translation 2 units up?

• Step 1 Multiply by 1 2 1 half  to compress.

  y = x 3 → y = 1 2 x 3 y equals , x cubed , rightwards arrow . y equals , 1 half , x cubed

• Step 2 Multiply by − 1 negative 1  to reflect.

  y = 1 2 x 3 → y = − 1 2 x 3 y equals , 1 half , x cubed , rightwards arrow y equals negative , 1 half , x cubed

  Think

  How is translating this cubic function like translating a quadratic function?

  In each case you replace x with x − h x minus h  to translate h units to the right.

• Step 3 Replace x with x − 3 x minus 3  to translate horizontally.

  y = − 1 2 x 3 → y = − 1 2 ( x − 3 ) 3 y equals negative , 1 half , x cubed , rightwards arrow y equals negative , 1 half . open , x minus 3 , close cubed

• Step 4 Add 2 to translate vertically.

  y = − 1 2 ( x − 3 ) 3 → y = − 1 2 ( x − 3 ) 3 + 2 y equals negative , 1 half . open , x minus 3 , close cubed . rightwards arrow y equals negative , 1 half . open , x minus 3 , close cubed . plus 2