Content

• To step

A tangent to a parabola or curve is a line that only touches the curve at a certain point.To find the equation of this tangent line, you will have to calculate the slope of the curve at that point, which requires a few mathematical calculations. You can then write the tangent equation in a point-slope form. This article explains which steps to take.

To step

1. The equation of a curve can be expressed as a function. Find the derivative of this function to find the equation of the slope of this curve.
   • The easiest way to differentiate most polynomials is through the chain rule. Multiply each equation of the function by its power to find that term's coefficient in the derivative, then reduce the power by 1.
   • Example: For the function f (x) = x ^ 3 + 2x ^ 2 + 5x + 1, is the derivative f "(x) = 3x ^ 2 + 4x + 5.
   • For f (x) = (2x + 5) ^ 10 + 2 * (4x + 3) ^ 5, the derivative is f '(x) = 10 * 2 * (2x + 5) ^ 9 + 2 * 5 * 4 * (4x + 3) ^ 4 = 20 * (2x + 5) ^ 9 + 40 * (4x + 3) ^ 4.
2. The coordinates where the tangent line touches the curve should be given. Enter the x value of this point into the derivative function to find the slope of the curve at that point.
   • For x = 2, it is the point on the curve (2,27) because f (2) = 2 ^ 3 + 2 * 2 ^ 2 + 5 * 2 + 1 = 27.
   • For f "(x) = 3x ^ 2 + 4x + 5, the slope is in (2,27) is f '(2) = 3 (2) ^ 2 + 4 (2) + 5 = 25.
3. This slope is also the slope of the tangent line. Now you have the slope and point of this line, so you can write the equation of the line in point-slope form, or y - y1 = m (x - x1).
   • In the point-slope form, is m the slope and (x1, y1) are the coordinates of the point. So in this example, the equation becomes y - 27 = 25 (x - 2).
4. You may also need to convert this equation to another form to get the final answer, should the problem instructions prompt you to do so.