Final answer:
To transform the graph of f(x) = x^2 into g(x) = 3(x + 2)^2 - 1, we apply a vertical stretch by a factor of 3 to make the graph narrower, shift it to the left by 2 units, and shift it down by 1 unit.
Step-by-step explanation:
Starting with the basic graph of f(x) = x^2, we need certain transformations to achieve g(x) = 3(x + 2)^2 - 1. Here are the steps we follow:
- The coefficient 3 in front of the squared term indicates a vertical stretch. This makes the graph of g(x) narrower than that of f(x) because it's stretched vertically by a factor of 3.
- The term (x + 2) inside the square indicates a horizontal shift to the left by 2 units. This is because the effect of adding 2 within the parentheses is to move the graph in the opposite direction of the addition.
- Lastly, the -1 at the end of the equation represents a vertical shift downward by 1 unit.
Putting it all together, g(x) is narrower than f(x), is shifted to the left by 2 units, and down by 1 unit.