Final answer:
The equation of the tangent to the curve y = 3x² - x at x = 1 is found by taking the derivative to determine the slope and then using the point-slope form of the line equation. The final equation of the tangent line is y = 5x - 3.
Step-by-step explanation:
To find the equation of the tangent to the graph y = 3x² - x at x = 1, we first need to determine the slope of the tangent line at that point. The slope of the tangent is given by the derivative of the function. Deriving the equation y = 3x² - x with respect to x, we get dy/dx = 6x - 1.
Substituting x = 1 into the derivative, we get the slope m = 6(1) - 1 = 5. The point on the curve at x = 1 can be found by substituting x = 1 back into the original equation to get y = 3(1)² - 1 = 2. So, the point of tangency is (1, 2).
The equation of the tangent line can be written in point-slope form, which is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point of tangency.
Substituting the known values, we get y - 2 = 5(x - 1). Expanding and simplifying, the equation of the tangent line is y = 5x - 3.