Given f(x)=x^2, describe (in words) the transformation on f that would produce g(x)=-2(x+4)^2-7?
To describe the transformation on f(x) that would produce g(x) = -2(x+4)^2 - 7, we need to analyze each part separately.
1. Inside the parentheses, we have (x + 4). This indicates a horizontal transformation since it affects the input value of the function. In this case, it means we will shift the graph of f(x) four units to the left.
2. Next, we square the expression (x + 4), which corresponds to the "x^2" term in f(x). This squared term preserves the shape of the graph, but it causes it to become narrower or wider depending on the coefficient. In this case, the coefficient is -2, so the graph will be reflected and become wider vertically.
3. After applying the square term, we multiply the result by -2, which further reflects the graph. This negative coefficient will flip it upside down.
4. Finally, we subtract 7 from the result, which corresponds to the "-7" term in g(x). This term will shift the entire graph seven units downward.
In summary, the transformation consists of a horizontal shift four units to the left, a vertical reflection, a vertical stretch by a factor of 2, and a vertical shift seven units downward, in that order.