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How to Find the Equation of the Tangent Line to a Graph

A common application of calculus is determining the equation of a line that intercepts a curve at a given point. This is called the tangent to a curve. As far as calculus problems are concerned, the good news is that this is one of the simplest concepts to grasp. The more advanced concepts build up on these basic rules. So, having a solid understanding of this process will set you up for success as you advance in your study of mathematics.

Instructions

- 1
  Assume an example curve equation of f(x) = x^2 + 3x - 4, where the notation 2x^2 is x squared. In our example we want the tangent at the point x = 4.
- 2
  Apply the calculus rule that the equation of tangent line to a given curve f(x) is y = f'(t)(x-t) + f(t) where t is the x coordinate of the tangent. To find f'(x), we need to compute the derivative of the curve's equation, which becomes f'(x) = 2x + 3
- 3
  Compute the slope using the f'(x) = 2x + 3 from Step 2 by substituting our example tangent point of x = 4. f'(4) = (2 * 4) + 3 = 8 + 3 = 11.
- 4
  Compute f(4) = (4*4) + (3*4) - 4 = 16 + 12 - 4 = 24
- 5
  Apply the values we just computed in Steps 3 and 4 back into the formula: y = f'(t)(x-t) + f(t). y = 11(x - 4) + 24 = 11x - 20. Thus the equation of the tangent line that intersects the curve f(x) = x^2 + 3x - 4 at x = 4 is y = 11x - 20.

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