1. What is a Rational Function? A rational function is a function of the form where p (x) and q (x) are polynomials and q (x)  0. p (x)q (x)p (x)q (x) f (x) = For example - y =

2. Graphing a Simple Rational Function A rational function is a function of the form where p (x) and q (x) are polynomials and q (x)  0. p (x)q (x)p (x)q (x) f (x) = For instance consider the following rational function: 1x1x y = The graph of this function is called a hyperbola and is shown below. Notice that the graph never crosses the x or y axes. An imaginary line that the graph approaches but usually do not cross is called an asymptote.

5. Graphing a Simple Rational Function 1414 1313 1212 1212 – 4 –2 –1 – 2 – 3 xy – – – – 4 2 1 1 2 3 xy 1414 1313 1212 1212 The x -axis is a horizontal asymptote. The y -axis is a vertical asymptote. The domain and range are all nonzero real numbers. The graph has two symmetrical parts called branches. For each point ( x, y ) on one branch, there is a corresponding point ( –x, –y ) on the other branch. Notice the following properties for y =

6. G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY Let p (x ) and q (x ) be polynomials with no common factors other than 1.The graph of the rational function has the following characteristics. f (x ) = = a m x m + a m – 1 x m – 1 + … + a 1 x + a 0 b n x n + b n – 1 x n – 1 + … + b 1 x + b 0 p (x )q (x )p (x )q (x ) 1. x - intercepts are the real zeros of p (x ) or when p(x) = 0 2. vertical asymptote at each real zero of q (x ) or when q(x) = 0

7. 3. at most one horizontal asymptote G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY f (x ) = = a m x m + a m – 1 x m – 1 + … + a 1 x + a 0 b n x n + b n – 1 x n – 1 + … + b 1 x + b 0 p (x )q (x )p (x )q (x ) To find horizontal asymptotes use the following procedure

8. G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY If upper > lower, the graph has no horizontal asymptote. It has a slant asymptote if the difference is 1. Find it by long division. If upper < lower, the line y = 0 is the horizontal asymptote. If upper = lower, the line y = is the horizontal asymptote. A rational function has an y = Example

9. Graphing a Rational Function (m < n) The bell-shaped graph passes through (–3, 0.4), (– 1, 2), (0, 4), (1,2), and (3, 0.4). Graph y =. 4 x 2 + 1 S OLUTION The numerator has no zeros, so there is no x -intercept. The denominator has no real zeros, so there is no vertical asymptote. The degree of the numerator (0) is less than the degree of the denominator (2), so the line y = 0 (the x-axis) is a horizontal asymptote.

10. Graphing a Rational Function (m = n) Graph y =. 3x 2 x 2 – 4 The degree of the numerator (2) is equal to the degree of the denominator (2), so the horizontal asymptote is y = = 3. 3131 S OLUTION The numerator has 0 as its only zero, so the graph has one x -intercept at (0, 0). The denominator can be factored as (x + 2)(x – 2), so the denominator has zeros at 2 and – 2. This implies vertical asymptotes at x = – 2 and x = 2.

11. To the left of x = – 2 To the right of x = 2 Between x = – 2 and x = 2 Graphing a Rational Function (m = n) To draw the graph, plot points between and beyond vertical asymptotes. Graph y =. 3x 2 x 2 – 4 4 5.4 –1 0 –1 5.4 4 – 4 – 3 –1 0 1 3 4 xy xy

12. Graphing a Rational Function (m > n) Graph y =. x 2 – 2x – 3 x + 4 The degree of the numerator (2) is greater than the degree of the denominator (1), so there is no horizontal asymptote. Since the difference in degrees is 1, there is a slant asymptote. Find it by dividing the numerator by the denominator. S OLUTION The numerator can be factored as ( x – 3) and ( x + 1); the x -intercepts are 3 and –1. The only zero of the denominator is – 4, so the only vertical asymptote is x = – 4.

13. Graph y =. x 2 – 2x – 3 x + 4 21 The slant asymptote is at y = x - 6

14. Graphing a Rational Function (m > n) To draw the graph, plot points to the left and right of the vertical asymptote. To the left of x = – 4 To the right of x = – 4 Graph y =. x 2 – 2x – 3 x + 4 – 20.6 –19.2 – 22.5 2.5 – 0.75 – 0.5 0.63 2.1 –12 – 9 – 6 – 2 0 2 4 6 y xy xy

15. Graphing a Rational Function with your calculator Graph y =. x 2 – 2x – 3 x + 4 Y1=(x 2 – 2x – 3)/(x+ 4) Y2= Y3= Begin by entering the function into Y= Notice that the graph has a vertical line at x = -4. This line is not part of the graph – it is simply the graphing calculator’s attempt to connect the two parts of the graph. To remove this line you can go to MODE and choose DOT instead of CONNECTED

16. Click for Graph Practice Foss Mountain Design - Rational EquationsFoss Mountain Design - Rational Equations