Rational Equations and Inequalities
■ Rational Functions are useful models for
■Inverse variation relationships
■Proportion or ratio problems
■Relationships approaching limits
Model: Product Exchange
■Product exchange functions give the relationship between quantities of two
items that can be produced by the same factory. An oil refinery can produce
gasoline, heating oil, or a combination of the two . Data was collected for gas
versus heating oil production in 1000 gallon allotments
■Determine a model for the product-exchange of gas and heating oil.
■View the graph in the next slide and address these questions
■What functions have we studied that have the same concavity and decreasing trend as the data?
■What in the context of the problem indicates choosing one of these functions over the other?
Rational Function Model
■Serves as good model for data that approaches a fixed value or a limiting value since rational functions have asymptotes.
■Serves as a good model when the context is a ratio or comparison , such as gas versus heating oil
■How do we determine a rational model?
■Derive 5and Graphing calculators do not fit rational functions to data
■Estimate model using basic rational power function y = A/x. Will A > 1 or A < 1?
■Solution: Using graphing technology and eyeballing the curve of best fit we get
■Translate to start at (0,100)
Power Function Models
■Difficult to find actual model
■ Linearize the data (using Exercise 51 or 53)
■Determine from other problem context not provided
■Determine the heating oil production if the gas production is 15 of the 1,000 gallon allotments.
■How do we solve rational equations?
Solving Rational Equations
■Solve the resulting polynomial equation
■Check the solutions to be sure none of them are extraneous
■Use graphic methods if analytic methods fail
■Solve the following example
■Class Participation Activity
■Solve inequalities with higher degree polynomial, radical, absolute value, or rational expressions using the graphic method
■Either set to zero and write one related function
■Or write a related equation for each side of the
Solving Inequalities –
■Given any algebraic function y = f(x) we can transform it.
■a > 1 stretch, 0 <
a < 1 compress, a < 0 reflection in x-axis
■b > 0 translation left, b < 0 translation right
■c > 0 translation up, c < 0 translation down
the following function