Write down the equation for the parabola. If possible, reduce the equation, until you have an expression close to the standard form. Parabolas are second degree equations with standard form: Y =aX^2 + bX +c
Where Y, X are variables, and a, b, c are numeric constants.
For example, consider the equation:
Y = 3X^2 +5X -10
Apply the derivative to the function. The derivative (symbolized by "dy/dx") will provide the slope for a tangent line in any point of the parabola.
From the example:
Y = 3X^2 +5X -10
dy/dx = 6X +5
Write down the point in which you wish to find a tangent line. Since parabolas are second degree equations, there are no restrictions for the points of a parabola that can have a tangent line. In fact, every single point of the parabola has a tangent line.
From the example:
dy/dx = 6X +5
Assume you wish to find the slope of the tangent line at (1, -2), X=1:
dy/dx = 6(1) +5 = 11
Use the slope to find the equation to the tangent line. The line will have the form Y = mX+b where Y, X are variable, m is the slope, and b is a constant.
To find b, use the point given on the parabola and the slope.
Continuing the example:
Y = mX +b
-2 = (11)(1) +b
b = -13
Therefore, the tangent line equation at point (1, -2) will be:
Y = 11X -13