Answer
The statement, “Because I have two points to work with, I use the formula for slope, $\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}},$ to find the slope of the tangent line to the graph of a function $\left( a,f\left( a \right) \right)$.” does not make sense.
Work Step by Step
The slope of the tangent line to the graph of a function $y=f\left( x \right)$ at $\left( a,f\left( a \right) \right)$ is given by $\underset{h\to 0}{\mathop{\lim }}\,$ $\frac{f\left( a+h \right)-f\left( a \right)}{h}$, provided that this limit exists.
When two points are given, then the line formed is called the secant line and the slope is given by $\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
At the tangent line, there is only one point of tangency.
Also, the slope of the tangent line is evaluated at a single point, so the two points are not necessary to find the tangent line.
Thus, the statement does not make sense.
