Lesson 8.6 Introduction to Rational Functions To investigate basic transformations of the inverse variation function f(x)=1/x To model real-world data with rational functions To write rational expressions in lowest terms.

Recall that you learned about the inverse function y=1/x in chapter 2 Recall that you learned about the inverse function y=1/x in chapter 2. Look at the graph of this function on the coordinate plane. One part is in quadrant I Another part is in quadrant II As x-values get closer to zero, the graph gets closer to the y-axis. As x-values get farther from zero, the graph gets closer and closer to the x- axis. An asymptote is a line that the graph approaches. x=0 and y=0 are both asymptotes for this function.

I’m Trying to Be Rational In the first part you will explore transformations of the parent function y = 1/x. Graph this function on your calculator. Use what you have learned about transformations to predict what the graphs of these functions will look like. Graph each equation along with y = 1/x. Compare your graphs with your predictions. How can you tell where asymptotes occur on your calculator screen?

Without graphing, describe what the graphs of these linear functions will look like. Use the words linear, nonlinear, increasing, and decreasing. Define the domain and range. Give equations for the asymptotes. A function is an inverse variation, when the product of x and y is constant. Do you think the equations in this investigation are inverse variations? Explain. Write a function that has an asymptote at x = -2 and y = 1. Sketch its graph and describe its domain.

Functions you have been studying in this lesson are called rational functions. Rational functions model many real-world applications.

A salt solution is made from salt and water A salt solution is made from salt and water. A bottle contains 1 liter of 20% salt solution. This means that the concentration is 20% salt or 0.2 of the whole solution. Show what happens to the concentration as you add water to the bottle in half- liter amounts. Added amounts of water (L) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 4.5 5.0 Amount of salt (L) 0.2 Whole Solution (L) 5.5 6.0 Concentration of Salt 0.133 0.1 0.08 0.067 0.057 0.05 0.044 0.04 0.036 0.033

Find an equation that models the concentration of salt as you add water. As the amount of whole solution increases, the concentration of salt decreases, but the amount of salt stays the same. This is an inverse variation and the constant of variation is the amount of salt. Confirm by placing data in the graphing calculator and using the equation to see if it fits the data.

How much water should you add to get a 2.5% salt solution? You will need to add 7 liters of water to have a 2.5% salt solution.

Rational expression look similar to fractions, but include variables as well as numbers. A rational expression is in lowest terms when the numerator and denominator have no factors in common other than 1. Reduce these rational expressions. Place the original expression in y1 and the simplified expression in y2. Verify that table of values are the same.

Perform the indicated operation and reduce the results to lowest terms.