Question 1.1: The Tangent Line to a Parabola Find the slope and an equat......
The Tangent Line to a Parabola
Find the slope and an equation of the tangent line to the parabola y = x² at the point P = (2,4) .
Let Q = (x,x²), x ≠ 2 , be another point on the parabola. The slope of the line joining P and Q is as follows:
m=\frac{\mathrm{change\;in}\,y}{\mathrm{change\;in}\,x}=\frac{x^{2}-4}{x-2}=\frac{(x-2)(x+2)}{x-2}=x+2,x\neq2\,.
Geometrically, as the point Q approaches P, the line joining P and Q approaches the tangent line at P. Algebraically, as X approaches 2, the slope of the line joining P and Q approaches the slope of the tangent line at P. Hence you see that the slope of the tangent line is m = 2 + 2 = 4 . Symbolically, we represent this limit argument as follows:
\operatorname*{lim}_{x\to2}\frac{x^{2}-4}{x-2}=\operatorname*{lim}_{x\to2}(x+2)=4\,.
The equation of the tangent line to the parabola at (2,4) is y – 4 = 4(x – 2) , or y = 4x – 4.
The tangent line problem uses the concept of limits, a topic we will discuss in Lessons Four through Six.