Horizontal, slant and polynomial end behavior

Horizontal, slant and polynomial end behavior

Study how rational functions behave for large positive and negative inputs. This precalculus practice links degree comparison and leading coefficients with polynomial division, giving a coherent way to identify horizontal lines, slant lines and higher-degree curves that describe end behavior.

Questions include approach from above or below, exact quotient-and-remainder identities, crossings of an asymptote and the distinction between matching a leading term and approaching a curve with vanishing vertical difference. Single-select and multiselect tasks move from familiar degree rules to justification and parameter reasoning.

Before starting, review polynomial division, signs of powers and vertical asymptotes. You will not need calculus. Every formula can be analyzed by division or by separating a dominant term from a remainder that becomes small.

Use the explanations to check the entire claim: which curve is approached, what happens at each end and whether a finite crossing is possible. The problems deliberately distinguish distant behavior from excluded finite inputs. A line that describes the far ends need not be a barrier, and a leading term alone need not give the full slant or polynomial curve. This practice supports later rational graph reconstruction and interpretation of mathematical models.