6.5 Graphing rational functions

Ideas in this lesson

Reciprocal transformations

For a translated reciprocal function, the horizontal and vertical shifts locate the intersection of its asymptotes.

Domain and intercepts

Exclude original denominator zeros. Numerator zeros give horizontal intercepts only where the function is defined; evaluate at zero for a vertical intercept when permitted.

Holes

A canceled denominator factor creates a removable hole. Evaluate the simplified rule at the excluded input to find its missing output.

Vertical asymptotes

After canceling common factors, zeros of the remaining denominator give vertical asymptotes.

Horizontal asymptotes

Compare numerator and denominator degrees. Equal degrees use the ratio of leading coefficients; a lower numerator degree gives a zero horizontal asymptote.

Slant asymptotes

When the numerator degree is one greater than the denominator degree, polynomial division gives the slant asymptote.

Behavior near an asymptote

Determine the sign of the small denominator on each side of a vertical asymptote to describe one-sided behavior.

Construct complete reciprocal graphs using asymptotes, points on both sides, branch orientation, domain and range

construct complete reciprocal graphs using asymptotes, points on both sides, branch orientation, domain and range.

Construct complete linear-over-linear graphs, rewrite in translated reciprocal form and justify domain/range; explicitly distinguish degenerate constant-with-a-hole cases

construct complete linear-over-linear graphs, rewrite in translated reciprocal form and justify domain/range; explicitly distinguish degenerate constant-with-a-hole cases.

Construct higher-degree graphs without real poles and determine exact or justified numerical ranges, including symmetry and bounded output examples

construct higher-degree graphs without real poles and determine exact or justified numerical ranges, including symmetry and bounded output examples.

Construct graphs across multiple poles, repeated factors and holes; locate valid zeros and points on every branch and distinguish touching/crossing behavior

construct graphs across multiple poles, repeated factors and holes; locate valid zeros and points on every branch and distinguish touching/crossing behavior.

Describe higher polynomial end behavior using quotient/remainder and distinguish it from an exact asymptote statement

describe higher polynomial end behavior using quotient/remainder and distinguish it from an exact asymptote statement.

Recover or construct a rational rule from a graph, selected points, excluded inputs, range or asymptote requirements

recover or construct a rational rule from a graph, selected points, excluded inputs, range or asymptote requirements.

Select a sensible graphing window, corroborate with tables, recognize spurious joined branches and hidden holes, and explain numerical limitations

select a sensible graphing window, corroborate with tables, recognize spurious joined branches and hidden holes, and explain numerical limitations.

Build/read rational tables and graphs in context, predict targets, interpret range/asymptotes/trends and reject extrapolation beyond the model's domain

build/read rational tables and graphs in context, predict targets, interpret range/asymptotes/trends and reject extrapolation beyond the model's domain.

Eliminate a variable under a geometric constraint to build a rational objective, then use a graph/table to optimize cylinder, prism or thick-walled-container models; verify feasible dimensions and compare designs

eliminate a variable under a geometric constraint to build a rational objective, then use a graph/table to optimize cylinder, prism or thick-walled-container models; verify feasible dimensions and compare designs.