Question

Describe the transformations that take place on the parent function (f(x) = 3ˣ) to obtain the graph of (f(x) = 3⁻ˣ⁺¹⁻²).

Answer 1

The parent function f(x) = 3^x is transformed to f(x) = 3^{-x+1}-2 by reflecting it across the y-axis, shifting it one unit to the left, and then shifting it down by two units.

Step-by-step explanation:

To describe the transformations applied to the parent function f(x) = 3^x to obtain the graph of the function f(x) = 3^{-x+1}-2, we will follow a step-by-step process:

1. The negative sign in front of the x in the exponent indicates a reflection across the y-axis. Originally, the graph of 3^x increases as x increases, but after the reflection, it will decrease as x increases.
2. The +1 inside the exponent (to the right of x) indicates a horizontal shift of one unit to the left. This is because we effectively take x and subtract 1 before applying the exponentiation. So every point on the graph moves one unit to the left.
3. Lastly, the -2 at the end of the equation signifies a vertical shift downward by two units. Every point on the graph is moved down two units in the y direction.

Combining these transformations, the graph of f(x) = 3^x is first reflected across the y-axis, then shifted one unit to the left, followed by a shift downward by two units to arrive at the graph of f(x) = 3^{-x+1}-2.