1. Copyright © 2011 Pearson Education, Inc. Slide 4.1-1 4.1 Rational Functions and Graphs Rational Function A function f of the form p/q defined by where p(x) and q(x) are polynomials, with q(x)  0, is called a rational function. Examples

2. Copyright © 2011 Pearson Education, Inc. Slide 4.1-2 The simplest rational function – the reciprocal function 4.1 The Reciprocal Function

3. Copyright © 2011 Pearson Education, Inc. Slide 4.1-3 4.1 The Reciprocal Function

4. Copyright © 2011 Pearson Education, Inc. Slide 4.1-4 4.1 Transformations of the Reciprocal Function The graph of can be shifted, translated, and reflected. Example Graph SolutionThe expression can be written as Stretch vertically by a factor of 2 and reflect across the y-axis (or x-axis).

5. Copyright © 2011 Pearson Education, Inc. Slide 4.1-5 4.1 Graphing a Rational Function Example Graph Solution Rewrite y: The graph is shifted left 1 unit and stretched vertically by a factor of 2.

6. Copyright © 2011 Pearson Education, Inc. Slide 4.1-6 4.1 Mode and Window Choices for Calculator Graphs Non-decimal vs. Decimal Window –A non-decimal window (or connected mode) connects plotted points. –A decimal window (or dot mode) plots points without connecting the dots. Use a decimal window when plotting rational functions such as –If y is plotted using a non-decimal window, there would be a vertical line at x = –1, which is not part of the graph.

7. Copyright © 2011 Pearson Education, Inc. Slide 4.1-7 4.1 Mode and Window Choices for Calculator Graphs Illustration Note: See Table for the y-value at x = –1: y 1 = ERROR.

8. Copyright © 2011 Pearson Education, Inc. Slide 4.1-8 4.1 The Rational Function f (x) = 1/x 2

9. Copyright © 2011 Pearson Education, Inc. Slide 4.1-9 4.1 Graphing a Rational Function ExampleGraph Solution Vertical Asymptote: x = –2; Horizontal Asymptote: y = –1.