Final answer:
The slope of the tangent line to the function f(x) = -2x² + 9x at x = -2 is calculated by finding the function's derivative, f'(x) = -4x + 9, and evaluating it at x = -2, which gives us a slope of 17.
Step-by-step explanation:
To find the slope of the tangent line to the function f(x) = -2x² + 9x at x = -2, we need to calculate the derivative of f(x) which gives us the slope of the tangent line at any point x. The derivative, denoted as f'(x), is found by applying the power rule to each term of the function.
Step 1: Apply the power rule. The derivative of -2x² is -4x, and the derivative of 9x is 9.
Step 2: Combine the derivatives to get the overall derivative, f'(x) = -4x + 9.
Step 3: Plug in the value of x (-2) into the derivative to find the slope at that point. So, f'(-2) = -4(-2) + 9 = 8 + 9 = 17. Therefore, the slope of the tangent line at x = -2 is 17.