1. The problem asks to graph the function $f(x) = x$ after it has been vertically stretched by a factor of 2 and vertically shifted down by 3 units. 2. The original function is $f(x) = x$. 3. A vertical stretch by a factor of 2 means multiplying the function by 2: $$f(x) = 2x$$ 4. A vertical shift down by 3 units means subtracting 3 from the function: $$f(x) = 2x - 3$$ 5. So the transformed function is $$f(x) = 2x - 3$$. 6. This is a linear function with slope 2 and y-intercept -3. 7. To graph it, plot points such as: - When $x=0$, $f(0) = 2(0) - 3 = -3$ - When $x=1$, $f(1) = 2(1) - 3 = -1$ - When $x=2$, $f(2) = 2(2) - 3 = 1$ 8. The graph is a straight line passing through these points. Final answer: The function after transformation is $$f(x) = 2x - 3$$.