beginning with the graph of f(x) =x^(2), what transformations are needed to form g(x)=(1)/(2)(x+4)^(2)-3
To transform the graph of f(x) = x^2 to g(x) = (1/2)(x+4)^2 - 3, you would need to apply the following transformations:
1. Horizontal translation: The graph needs to be shifted 4 units to the left. This is achieved by changing f(x) to f(x+4).
2. Vertical translation: The graph needs to be shifted downwards 3 units. This is achieved by subtracting 3 from f(x).
3. Vertical scaling: The graph needs to be vertically compressed by a factor of 1/2. This is achieved by multiplying f(x) by 1/2.
Therefore, the transformations are:
- Horizontal translation: f(x) → f(x+4)
- Vertical translation: f(x+4) → f(x+4) - 3
- Vertical scaling: f(x+4) - 3 → (1/2)(f(x+4) - 3)
So, the final transformation is: f(x) → (1/2)(x+4)^2 - 3.