Chapter 4: Rational, Power, and Root Functions 4.1 Rational Functions and Graphs 4.2 More on Rational Functions and Graphs 4.3 Rational Equations, Inequalities, Models, and Applications 4.4 Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions

4.1 Rational Functions and Graphs 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

4.1 The Reciprocal Function The simplest rational function – the reciprocal function

4.1 The Reciprocal Function

4.1 Transformations of the Reciprocal Function The graph of can be shifted, translated, and reflected. Example Graph Solution The expression can be written as Stretch vertically by a factor of 2 and reflect across the y-axis (or x-axis).

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.

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.

4.1 Mode and Window Choices for Calculator Graphs Illustration Note: See Table for the y-value at x = –1: y1 = ERROR.

4.1 The Rational Function f (x) = 1/x2

4.1 Graphing a Rational Function Example Graph Solution Vertical Asymptote: x = –2; Horizontal Asymptote: y = –1.