Polynomials and Rational Functions (2.1)

1. Polynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial

2. Shapes of Polynomials • Look at the shape of the odd degree polynomials

3. Graph of Odd polynomial

4. Graph of Odd Polynomial

5. Graphs of even degree polynomials • Now, look at the shape of the even degree polynomial

6. Graph of even degree polynomial • Here is another example of an even degree polynomial :

7. Generalization: • The graphs of odd-degree polynomials start negative, end positive and cross the x-axis at least once. • The even-degree polynomial graphs start positive, end positive, and may not cross the x axis at all

8. Characteristics of polynomials: • Graphs of polynomials are continuous. One can sketch the graph without lifting up the pencil. • 2. Graphs of polynomials have no sharp corners. • 3. Graphs of polynomials usually have turning points, which is a point that separates an increasing portion of the graph from a decreasing portion.

9. Turning points and x intercepts • Theorem 1 : Turning points and x Intercepts of Polynomials • The graph of a polynomial function of positive degree n can have at most n-1turning points and can cross the x axis at most n times.

10. Largest value of the roots of a polynomial • Theorem 2: Maximum value of an x-intercept of a polynomial. If r is a zero of the polynomial P(x) this means that P(r) = 0. For example, • is a second degree polynomial . and • , so r = 4 is a zero of the polynomial as well as being an x-intercept of the graph of p(x).

11. Cauchy’s Theorem • A theorem by a French mathematician named Cauchy allows one to determine the maximum value of a zero of a polynomial (maximum value of the x-intercept). • Let’s take an example: the polynomial

12. Cauchy’s Theorem • According to this theorem • The numbers within the absolute value symbols are the coefficients of the polynomial p(x). < 1 + maximum value of = 1 + 4 = 5

13. Result of application of Cauchy’s theorem • From this result we have , which means -5 < r < 5 . This tells us that we should look for any potential x intercepts within the range of -5 and 5 on the x –axis. In other words, no intercepts (roots) will be found that are greater than 5 nor less than -5.

14. Conclusion , we find that the other zero is located at (0,0). Thus, the two zeros , 0 , -4, are within the range of -5 to 5 on the x-axis. Now, let’s try another example: • From the graph of

15. An Example: • Example: Approximate the real zeros of • First step: Coefficient of cubic term must equal one, so divide each term by three to get a new polynomial Q(x)= Roots of new polynomial are the same as the roots of P(x).

16. Example, continued • Step 2: Use the theorem:

17. Example, continued • Step3: We know that all possible x intercepts (roots) are found along the x-axis between -5 and 5. So we set our viewing rectangle on our calculator to this window and graph the polynomial function. • Step 4. Use the zero command on our calculator to determine that the root is approximately -3.19 (there is only one root).

18. Rational Functions • Definition: Rational function: a quotient of two polynomials, P(x) and Q(x), is a rational function for all x such that Q(x) is not equal to zero. Example: Let P(x) = x + 5 and Q(x) = x – 2 then R(x)= is a rational function that is defined for all real values of x with the exception of 2 (Why?)

19. Domain of rational functions • Domain : • and x is a real number. This is read as “the set of all numbers, x , such that x is not equal to 2. • X intercepts of a rational function: To determine the x-intercepts of the graph of any function, we find the values of x for which y = 0 . In our case y = 0 implies that 0 = • This implies that x + 5 = 0 or x = -5 .

20. Y-intercept of a rational function • Y intercept: The y intercept of a function is the value of y for which x = 0 . Setting x = 0 in the equation we have y = , or -5/2. So, the y-intercept is located at ( 0, -2.5). Notice that the y-intercept is a point described by an ordered pair, not just a single number. Also, remember that a function can have only one y intercept but more than one x-intercept • ( Why?)

21. Graph of a Rational function: • Plot points near the value at which the function is undefined. In our case, that would be near x = 2. Plot values such as 1.5, 1.7. 1.9 and 2.1, 2.3, 2.5. Use your calculator to evaluate function values and make a table. • Determine what happens to the graph of f(x) if x increases or decreases without bound. That is, for x approaching positive infinity or x approaching negative infinity. • Sketch a graph of a function through these points. 4. Confirm the results using a calculator and a proper viewing rectangle.

22. Graph of rational function

23. Conclusions: • From the graph we see that there is a vertical asymptote at x = 2 because the graph approaches extremely large numbers as x approaches 2 from either side. • We also see that y = 0 is a horizontal asymptote of the function since y tends to go to zero as x tends to either a very large positive number or very small negative number.