Identify the degree of the polynomial, leading coefficient, roots (x-intercepts), y-intercept, and end behavior.
An even degree polynomial has both ends pointing in the same direction (upward if the leading coefficient is positive, downward if negative). An odd degree polynomial has ends pointing in opposite directions.
The roots indicate where the graph crosses or touches the x-axis. The multiplicity of each root determines whether the graph crosses or touches the x-axis at that point.
The y-intercept is found by setting x = 0 in the polynomial function and solving for y.
If a root has an odd multiplicity, the graph crosses the x-axis at that root. If a root has an even multiplicity, the graph touches the x-axis and turns around at that root.
Choose x-values between the roots and evaluate the function to determine whether the graph is above or below the x-axis in those intervals.
A positive leading coefficient means the graph rises to the right, while a negative leading coefficient means the graph falls to the right.
Focus on the key features: end behavior, intercepts, and behavior at roots. Use these to create a rough sketch, and refine as needed.