The equation for the vertically stretched graph g(x) is 5 * 7^x, and the correct answer is (D).
To vertically stretch the graph of f(x) = 7^x by a factor of five, we modify the function to g(x) = 5 * 7^x. The general form for a vertical stretch or compression is a f(x), where a is the vertical scaling factor. In this case, a = 5 indicates a vertical stretch.
Each y-coordinate in the new graph, g(x), is obtained by multiplying the corresponding y-coordinate in f(x) by 5. This means that for every x, g(x) is five times larger than f(x). The base of the exponential function, 7^x, remains unchanged, as the stretching only affects the output values.d
Option (D) g(x) = 5 * 7^x correctly captures this vertical stretching, making it the appropriate choice. Options (A), (B), and (C) involve incorrect combinations of the base and scaling factor, and thus, do not represent a vertical stretch of f(x).