Tangent+lines+vs.+Normal+lines

__Tangent Lines vs Normal Lines__
Tangent lines touch a curve at one evaluated point but can intersect the graph at other points as well. This line is produced by taking a derivative to find slope and utilizing a given point of intersection to find the intercepts. The intercepts and slope can then be used to write the equation of the line.

Normal Lines are the lines perpendicular to the tangent lines, but still pass through the original evaluation point.

Example: Find the tangent line and normal line to the graph y=3x 2 +4x+3 at (0,3)

Solution: y'=6x+4 by the power rule. y'(0)=4 so the slope of the tangent line at x=0 is 4, now point slope can be utilized: (y-y 1 )=m(x-x 1 ), where (x 1 ,y 1 ) is the original coordinate and m is the slope. (y-3)=4(x-0)=4x This can be changed to slope intercept form Now time to find the normal line y=(-1/4)x+b, as perpendicular lines have a slope equal to the negative reciproal of the original Plug in the original point to solve for b: 3=(-1/4)(0)+b=b Yielding: **y=(-1/4)x+3** as the Normal Line equation.
 * y=4x+3**, this is the equation of the tangent line