# Curve Sketching:

## An 8-step menu for polynomial and rational functions

- Asymptotes
- Vertical: The values of x for which the given function is not defined.
- Form:
- Horizontal: The values of y which the function approaches as x becomes very large.
- Form:
- Oblique: The values of y which the function approaches as x becomes very large
- Form:

- Domain and Range
- Domain: The values of x used to generate the curve.
- The values of x which are not vertical asymptotes.
- Range: The values of y used to generate the curve.
- The values of y which are not horizontal asymptotes.

- Symmetry
- Corresponding value of x and y reflected about the x-axis, y-axis, or the origin.
- Symmetry about x-axis: let and test it.
- Symmetry about y-axis: let and test it.
- Symmetry about the origin: let and then test it.

- First Derivative
- We obtain Critical Points and examine the behavior of the curve. We determine the intervals along the x-axis where the curve increases and decreases by examining the slopes of the tangent lines.

- Second Derivative
- Maxima/Minima: We use the 2
^{nd} Derivative Test to determine which of the critical points are maxima or minima.
- Points of Inflection: The values of x where the curve changes concavity i.e. the values of x where the curve stops being concave down and begins being concave up (and vice versa).
- Concavity: We examine the intervals along the x-axis where the curve is either concave up or down.

- Intercepts
- x-intercepts: The values of x where the function intercepts the x-axis.
- Let and solve for x.
- y-intercepts: The values of y where the function intercepts the y-axis.
- Let and solve for y.

- y-values
- The corresponding values of y for the maxima, minima, and points of inflection.

- Sketch