Final answer:
To transform the parent function f(x) = 3^x into f(x) = 3^-x + 1, you need to reflect the graph across the y-axis and then translate it upward by 1 unit. This changes the graph from a growth to a decay function and shifts the horizontal asymptote from y=0 to y=1. Common mistakes include reflecting across the wrong axis and misordering transformations.
Step-by-step explanation:
To transform the parent function f(x) = 3^x to obtain f(x) = 3^-x + 1, follow these steps:
- Apply a reflection transformation across the y-axis by replacing x with -x, resulting in the function 3^-x. This inverts the input which results in the graph flipping over the y-axis.
- Add 1 to the entire function, f(x) = 3^-x + 1, to achieve a vertical translation. This raises the graph by one unit in the y-direction.
Graphically, we start with the parent function and first reflect it across the y-axis, and then shift it upwards by 1 unit. The base function 3^x is a growth function that increases rapidly, but once reflected to 3^-x, it becomes a decay function which diminishes as x increases. Finally, by adding 1, every point is shifted one unit up, creating a new horizontal asymptote at y = 1 rather than at y = 0.
Common mistakes to avoid during this transformation include mixing up the reflection across the x-axis instead of the y-axis, forgetting to shift the graph vertically, or adding 1 before applying the reflection transformation which would alter the graph incorrectly.