Tangent Line to a Function

Finding the tangent line to the graph of a function at a single point can be extremely useful when interpreting the information that the function represents. So first to describe what a tangent line is: A tangent line of a function at one point shows the direction that the function is going at that point (Fig. 1). Theoretically the tangent line is only touching the curve of the function at one single point, or the point of tangency. To find the equation of the tangent line, certain bits of information are required.
One of these bits of information required is the slope of the tangent line. To find the slope of the tangent line of a function at a single point, the equation is used, assuming that “a” is the single point on the equation. The rest of this paper will be used to describe, through graphical methods, why this equation finds the slope of the tangent line. The slope of any linear equation can be described as rise over run, y over x, the output of a function over the input of a function, or the dependent variable over the independent variable.
All of these terms mean the same thing: the Y value on a graph over the X value on the graph. If the equation is examined closely, then it is clear that it represents a slope. The equation has the change of two output values, g(x) – g(a), over the change of two input values, x – a. The equation uses the change of an output, and the change of an input because two points on the graph is the minimum amount of information required to create a line. Fig. 2 and Fig. show how the two points on a graph can create an accurate tangent line. Fig. 2 shows that two points on the function can create a secant line with a slope that is approximately close to the slope of the tangent line, but it is not accurate enough. Fig. 3 shows that as the second point, D, on the function moves closer to the original point, C, the slope of the secant line approaches the slope of the tangent line. This movement shows how the slope of the secant line is equal to the equation.

All the equation for the slope of the secant line is the change in the Y value over the change of the X value. As point D gets closer to point C, the reason why finding the tangent line has to be a limit equation, and not just the secant line equation, becomes clear. The denominator of the secant slope function makes it so x cannot equal a. If x were to equal a, then the equation would be undefined because the denominator cannot equal 0. So the slope of the tangent line is the limit as D approaches C.