To find the function for g(x) given that g(x) is f(x) stretched vertically by a factor of 3, translated 2 units to the left, and 8 units up, we can follow these steps:
1. Start with the original function f(x) = x^2.
2. Stretch the function vertically by a factor of 3:
- Multiply the y-values of f(x) by 3.
- The function becomes f(x) = 3x^2.
3. Translate the function 2 units to the left:
- Replace x with (x + 2) in the function.
- The function becomes f(x) = 3(x + 2)^2.
4. Translate the function 8 units up:
- Add 8 to the y-values of the function.
- The final function is g(x) = 3(x + 2)^2 + 8.
By following these steps, we have determined the function for g(x) based on the given transformations applied to the original function f(x) = x^2. The final function for g(x) is g(x) = 3(x + 2)^2 + 8.
I hope this explanation clarifies how to find the function for g(x) based on the given transformations. Let me know if you have any further questions.