Final answer:
To find the equation of the tangent line to the function f(x) at point (1, -1), calculate its derivative to get the slope at that point. The slope of the tangent line at (1, -1) is found to be -7. Thus, the equation of the tangent line is y = -7x + 6.
Step-by-step explanation:
To find the equation of the line tangent to the graph of a function at a given point, you first need to find the slope of the tangent line at that point. The slope of the tangent line is the value of the derivative of the function at that point. We have the function f(x) = x(1 - 2x)^3.
First, let's find the derivative of f(x). Using the product rule, the derivative is f'(x) = (1 - 2x)^3 + x(3)(1 - 2x)^2(-2). To find the slope of the tangent at the point (1, -1), we plug in x = 1 into the derivative: f'(1) = (1 - 2*1)^3 + 1*3(1 - 2*1)^2(-2) = -1 - 6 = -7. Thus, the slope of the tangent line at (1, -1) is -7.
With the slope and the point (1, -1) known, we can use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point the line passes through. Substituting the slope and the point gives us the equation of the tangent line: y - (-1) = -7(x - 1), which simplifies to y = -7x + 6.