Find the derivative of the parabola as a first step toward finding the tangent line at a point. The functions for all parabolas are polynomials, and the derivatives are found by applying, term by term, using the pattern: aX^n becomes anX^(n - 1). This means that the derivative of X^3 - 5X^2 +3X -11 is 3X^2 - 10X + 3. Notice that the derivative of a constant term is always zero. The derivative describes how a function changes, and constants do not change.
Compute the slope at a particular point by plugging the X coordinate of the parabola into the derivative --- this will also give the slope of the tangent line at that point. For example, consider the parabola formed by the equation Y = X^2 --- an upward opening parabola with vertex at (0,0). The point (1,1) is on the parabola because 1 = 1^2, which fits the formula Y = X^2. The derivative of X^2 is 2X, so the slope of the parabola at (1,1) is 2(1) = 2. Now that you know the slope and one point, you can find the formula of the tangent line.
Use the point slope formula to find the equation of the tangent line. The formula is Y -Y1 = m(X - X1), where "m" is the slope and (X1,Y1) is the point. The line tangent to the parabola Y = X^2 at the point (1,1) is given by the formula Y - 1 = 2(X - 1) or Y = 2X -1. Another point on this parabola is (2,4), and the slope at this point is 2X = 2(2) = 4. The tangent line at this point is given by the formula Y - 4 = 4(X - 2) or Y = 4X - 4.