Exercise 36 Page 967

We want to find the exponential equation of the graph of y=2^x after reflecting it across the y-axis and moving it down1 unit. To do so, let's approach one transformation at a time.

Reflection

First, we want to reflect y = 2^x across the y-axis. To do it, let's review the three main types of reflections on a coordinate plane.

1. A reflection in the x -axis changes the sign of the y-variable: (x,y) → (x, - y).
2. A reflection in the y -axis changes the sign of the x-variable: (x,y) → (- x, y).
3. A reflection in the line y=x interchanges the x- and y-variables: (x,y) → (y,x).

   Therefore, for the indicated reflection, we have to change the sign of the x-variable. y=2^(-x)

   Translation

   Next, we want move y=2^(- x) down1 unit. To do that, we should recall two things.

   • If a horizontal translation is to the right, we subtract from the x-variable. If the translation is to the left, we add to the x-variable.
   • If a vertical translation is up, we add to the y-variable. If the translation is down, we subtract from the y-variable.

   This means that for the indicated translation, we have to subtract1 from the entire function. y=2^(-x) - 1 This corresponds to option C.