2. 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

3. 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. 4.1 The Reciprocal Function
The simplest rational function – the reciprocal function

5. 4.1 The Reciprocal Function

6. 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).

7. 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.

8. 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.

9. 4.1 Mode and Window Choices for Calculator Graphs
Illustration
Note: See Table for the y-value at x = –1: y1 = ERROR.

10. 4.1 The Rational Function f (x) = 1/x2

11. 4.1 Graphing a Rational Function
Example Graph
Solution
Vertical Asymptote: x = –2;
Horizontal Asymptote: y = –1.