Using the knowledge that we know from our sections on limits, continuity, first derivatives, and second derivatives, we are able to proceed with graphing functions. The process that we undergo to do so is referred to as curve sketching.
The following factors are crucial when sketching a graph:
Critical points
Points of inflection
ex. \( f(x) = 2x^4 + x^3 \)
\( f'(x) = 8x^3 + 3x^2 \)
\( f''(x) = 24x^2 + 6x \Longrightarrow 0 = 6x( 4x + 1) \Longrightarrow x = 0, -\frac{1}{4} \) are potential points of inflection of \( f(x) \)
Each of these factors will aid us in determining the graph's general shape, direction and behaviour.
Every function is different, and an understanding of each factor is very important. We will consider two examples and view the approach in sketching their curves.
Consider \( f(x) = (x-1)^2(x^2 - 16) \) and \( g(x) = \frac{x^2 - x - 2}{x+ 1}\). Use curve sketching methods to draw both of their graphs on the Cartesian Plane.
See the videos below for the solutions: