Final answer:
To find the equation of the normal line to the parabola y = x² - 8x + 1, find the derivative and the slope of the tangent line. Then, find the negative reciprocal of the slope to get the slope of the normal line. Finally, use the point-slope form of a line to write the equation of the normal line.
Step-by-step explanation:
The equation of a normal line to a parabola can be found by finding the derivative of the equation of the parabola, and then finding the negative reciprocal of the slope of the tangent line at the point of tangency.
Given the parabola equation y = x² - 8x + 1, we first find the derivative: y' = 2x - 8.
Next, we find the slope of the tangent line at a specific point by substituting the x-coordinate of the point into the derivative. Let's say we want to find the normal line at x = 3. Substitute x = 3 into the derivative: y' = 2(3) - 8 = -2. This is the slope of the tangent line.
The negative reciprocal of -2 is 1/2. So, the slope of the normal line is 1/2. Now, we can use the point-slope form of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope of the line. Substituting x₁ = 3, y₁ = 3² - 8(3) + 1 = -3, and m = 1/2, we get the equation of the normal line as y + 3 = 1/2(x - 3).