The tangent line to y = f(x) at a point (a, f(a) ) has slope dy/dx at x = a. So first compute the derivative:
y = x² - 9x → dy/dx = 2x - 9
When x = 4, the function takes on a value of
y = 4² - 9•4 = -20
and the derivative is
dy/dx (4) = 2•4 - 9 = -1
Then use the point-slope formula to get the equation of the tangent line:
y - (-20) = -1 (x - 4)
y + 20 = -x + 4
y = -x - 24
The normal line is perpendicular to the tangent, so its slope is -1/(-1) = 1. It passes through the same point, so its equation is
y - (-20) = 1 (x - 4)
y + 20 = x - 4
y = x - 24
