1. Introduction Though we may be unaware of the role they play, rational functions are an important part of daily life. Picture a patient in a doctor’s office who needs to be put on blood pressure medication. How does the doctor know the correct dosage of medication, and how frequently it should be taken, for that person? Or, imagine a factory that manufactures bicycles. How does the production manager determine the optimal amount of bicycles to produce that will yield the highest amount of profit and yet minimize production costs? Across many fields, rational functions are used regularly to help make important decisions. 1 3.3.1: Creating Rational Equations in Two Variables

2. Key Concepts A rational function is the quotient of two polynomial functions. In mathematical terms, the rational function f(x) is defined as, where p(x) and q(x) are polynomial functions and q(x) ≠ 0. For example, the rational function consists of the polynomial functions g(x) = 2x – 1 and h(x) = x 2 – 4. 2 3.3.1: Creating Rational Equations in Two Variables

3. Key Concepts, continued Recall that the domain of a function is the set of all input values (x-values) that satisfy the given function without restriction. The degree of a polynomial is the greatest exponent attached to the variables in the polynomial. The domain of a first-degree rational function is found by identifying the asymptotes. An asymptote is a line that a function gets closer and closer to, but never crosses or touches. A vertical asymptote is a straight line of the form x = b which the graph of a rational function approaches but never touches. 3 3.3.1: Creating Rational Equations in Two Variables

4. Key Concepts, continued A horizontal asymptote is a straight line of the form y = a that occurs at the maximum or minimum of the graph of a rational function. The domain will contain every real number except for the point at which the vertical asymptote falls. Recall that the domain of a function can be expressed in interval notation. That is, the domain is written in the form of (a, b), where a and b are the endpoints of the interval. Depending on the values of the interval, the notation may change. 4 3.3.1: Creating Rational Equations in Two Variables

5. Key Concepts, continued The table that follows summarizes how to note the intervals for various situations. Notice that a parenthesis next to a number indicates that the number is not included in the solution set; however, a bracket indicates that the number is part of the solution set. 5 3.3.1: Creating Rational Equations in Two Variables

6. Key Concepts, continued 6 3.3.1: Creating Rational Equations in Two Variables Interval notation ExampleDescription (a, b)(2, 10) All numbers between 2 and 10; endpoints are not included. [a, b][2, 10] All numbers between 2 and 10; endpoints are included. (a, b](2, 10] All numbers between 2 and 10; 2 is not included, but 10 is included. [a, b)[2, 10) All numbers between 2 and 10; 2 is included, but 10 is not included.

7. Key Concepts, continued The zeros of a rational function are the x-intercepts of the function; in other words, the domain value(s) for which f(x) = 0. Zeros are also known as roots. In the example, the domain values are found by assuming that the denominator x 2 – 4 ≠ 0. This means that the numerator 2x – 1 = 0 if f(x) = 0. Therefore, the zero for this function is found by solving the equation 2x – 1 = 0 for x. The result is. Thus, the graph intersects the x-axis at the point. 7 3.3.1: Creating Rational Equations in Two Variables

8. Key Concepts, continued The values of the domain for which a rational function is undefined are found by determining the values of the domain for which the denominator is equal to 0. In the example the values of the domain for which the function is undefined are found by solving the equation x 2 – 4 = 0 for x, which gives x = ±2. Therefore, the function is undefined at the domain values of x = 2 and x = –2. 8 3.3.1: Creating Rational Equations in Two Variables

9. Key Concepts, continued The values of the domain at which a rational function is undefined are also the equations of its vertical asymptotes. The vertical asymptotes do not include the points that make the function true. 9 3.3.1: Creating Rational Equations in Two Variables

10. Key Concepts, continued In the example,, the asymptotes are x = 2 and x = –2, because, which is undefined, and, which is also undefined. 10 3.3.1: Creating Rational Equations in Two Variables

11. Key Concepts, continued Another way to identify the asymptotes and zeros of a rational function is to look at a table of values generated on a graphing calculator. Follow the steps appropriate to your calculator model, beginning on the next slide. 11 3.3.1: Creating Rational Equations in Two Variables

12. Key Concepts, continued On a TI-83/84: Step 1: Press [Y=]. Press [CLEAR] to delete any other functions stored on the screen. Step 2: At Y 1, use your keypad to enter values for the function. Use [X, T, θ, n] for x and [x 2 ] for any exponents. Step 3: To view a table of values for the function, press [2ND][GRAPH]. Step 4: Arrow up and down the list of ordered pairs to find the zeros and the domain values for which the function is undefined. 12 3.3.1: Creating Rational Equations in Two Variables

13. Key Concepts, continued On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step 2: Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function to the right of “f1(x) =” and press [enter]. Step 4: To see a table of values, press [menu], then use the arrow keys to select 2: View, then 9: Show Table. Step 5: Arrow up and down the list of ordered pairs to find the zeros and the domain values for which the function is undefined. 13 3.3.1: Creating Rational Equations in Two Variables

14. Common Errors/Misconceptions identifying a function as a rational function when it is not trying to identify the zero of a function by setting x = 0 finding f(x) = 0 when the denominator or the numerator and denominator of the rational function is 0 writing the horizontal asymptote(s) of a rational function in the form x = a writing the vertical asymptote(s) of a rational function in the form y = b 14 3.3.1: Creating Rational Equations in Two Variables

15. Guided Practice Example 3 Write a rational function that has a zero of x = –1 and is undefined at x = 2. 15 3.3.1: Creating Rational Equations in Two Variables

16. Guided Practice: Example 3, continued 1.Use the definition of a rational function to write the simplest general polynomials for the numerator and the denominator. Recall the definition of a rational function is. The simplest polynomials for the numerator and the denominator have the form f(x) = x + a and g(x) = x + b. Therefore,. 16 3.3.1: Creating Rational Equations in Two Variables

17. Guided Practice: Example 3, continued 2.Determine the polynomial for the numerator. Use the zero of the function to determine the numerator. We are given that a zero of this function is at x = –1; therefore, f(–1) = 0. 17 3.3.1: Creating Rational Equations in Two Variables f(x) = x + aNumerator of the rational function (0) = (–1) + aSubstitute 0 for f(x) and –1 for x. a = 1Add 1 to each side to solve for a.

18. Guided Practice: Example 3, continued Now that we know a = 1, substitute 1 for a in the numerator of the rational function. Therefore, f(x) = x + 1, and x + 1 is the polynomial for the numerator. 18 3.3.1: Creating Rational Equations in Two Variables f(x) = x + aNumerator of the rational function f(x) = x + (1)Substitute 1 for a. f(x) = x + 1Simplify.

19. Guided Practice: Example 3, continued 3.Determine the polynomial for the denominator. The denominator is undefined for values that would result in a 0 in the denominator. We are given that the function is undefined at x = 2; therefore, g(2) = 0. 19 3.3.1: Creating Rational Equations in Two Variables g(x) = x + bDenominator of the rational function (0) = (2) = bSubstitute 0 for g(x) and 2 for x. b = –2 Subtract 2 from each side to solve for b.

20. Guided Practice: Example 3, continued Now that we know b = –2, substitute –2 for b in the denominator of the rational function. Therefore, g(x) = x – 2, and x – 2 is the polynomial for the denominator. 20 3.3.1: Creating Rational Equations in Two Variables g(x) = x + b Denominator of the rational function g(x) = x + (–2)Substitute –2 for b. g(x) = x – 2Simplify.

21. Guided Practice: Example 3, continued 4.Write the rational function using the polynomials determined for the numerator and denominator. Substitute the functions determined in steps 2 and 3 for the numerator and denominator of the definition of a rational function. 21 3.3.1: Creating Rational Equations in Two Variables Definition of a rational function Substitute x + 1 for f(x) and x – 2 for g(x).

22. Guided Practice: Example 3, continued This rational function is already in simplest form; therefore, the rational function,, has a zero of x = –1 and is undefined at x = 2. 22 3.3.1: Creating Rational Equations in Two Variables ✔

23. Guided Practice: Example 3, continued 23 3.3.1: Creating Rational Equations in Two Variables

24. Guided Practice Example 4 Identify the vertical asymptote(s) for the rational function 24 3.3.1: Creating Rational Equations in Two Variables

25. Guided Practice: Example 4, continued 1.Identify the values of x for which the rational function is undefined. The rational function is undefined for values that would result in a 0 in the denominator. Therefore, any factors in the denominator (x – 2, x + 3, and x – 4) that are equal to 0 would result in a value of that gives an undefined value of f(x). 25 3.3.1: Creating Rational Equations in Two Variables

26. Guided Practice: Example 4, continued Set each factor equal to 0 and solve the equation for to determine each value that makes f(x) undefined. The values of x = 2, x = –3, and x = 4 result in an undefined rational function. 26 3.3.1: Creating Rational Equations in Two Variables x – 2 = 0x + 3 = 0x – 4 = 0 x = 2x = –3x = 4

27. Guided Practice: Example 4, continued 2.List the vertical asymptotes for the rational function. The vertical asymptotes are located at the values of x for which the function is undefined and are of the form x = a. Therefore, the vertical asymptotes of the rational function,, are x = 2, x = –3, and x = 4. 27 3.3.1: Creating Rational Equations in Two Variables ✔

28. Guided Practice: Example 4, continued 28 3.3.1: Creating Rational Equations in Two Variables