Chapter 12 Rational Expressions and Functions 12 – 1 Inverse Variation If you are looking at the relationship between two things and one increases as the other decreases, then the relationship is known as inverse variation Some ways to word the inverse variation relationship: 1) y varies inversely with x 2) y varies inversely as x 3) y is inversely proportional to x Equations for inverse variation: xy = k or y = k/x where k cannot be 0 (k is called the constant of variation)

If you are given one point in an inverse variation relationship, you can find the constant of variation and write the equation Ex1. Suppose y varies inversely with x. If y = 12 and x = 3, write an equation for the inverse variation. Ex2. The points (2, 5) and (6, y) are two points on the graph of an inverse variation. Find the missing value. Inverse variation (along with direct variation) is often used in physics (see ex. 3 on page 638)

To check if a relationship is direct variation or inverse variation, test to see if xy is always the same value or if y/x is always the same value If xy is constantly the same, then it is inverse variation If y/x is constantly the same, then it is direct variation See example 4 on page 639

12 – 2 Graphing Rational Functions In simplest form, a rational function has a polynomial of at least degree 1 in the denominator Inverse variation relationships are rational functions Rational functions (and several other types of functions) have asymptotes Asymptotes are lines that the graph continually approaches, but never touches

Asymptotes can be vertical or horizontal (rational functions have both) A vertical asymptote can be found at whatever value would make the denominator equal zero Open to page 645 for vertical asymptotes and 646 for horizontal asymptotes The graph of a rational function in the form has a vertical asymptote at x = b and a horizontal asymptote at y = c. The graph is a translation of b units right or left and c units up or down

Be familiar with the families of functions on page 647 (equations and shape of graphs) When you are drawing the graphs, you must have the general shape completely correct as well as the location of the asymptotes

12 – 3 Simplifying Rational Expressions Any expression that has a variable in the denominator is a rational expression A rational expression is in simplest form if the numerator and the denominator have no common factors except 1 To simplify rational expressions, you will often have to factor (chapter 9) Ex1. Simplify

Simplify each of the following Ex2. Ex3.

12 – 4 Multiplying and Dividing Rational Expressions You multiply rational expressions like you do rational numbers Be sure to reduce if possible Multiply and simplify (if possible). Leave in factored form. Ex1. Ex2.

Divide rational expressions just like you would rational numbers Leave answers in factored form Remember to flip the 2 nd rational expression Divide Ex3. Ex4.

12 – 5 Dividing Polynomials To divide a polynomial by a monomial, divide each term of the polynomial by the monomial divisor (you will often end up with rational parts to your function) Ex1. Divide. To divide a polynomial by a binomial, you follow the same process you use in long division If the dividend has terms missing (i.e. x³ + x + 1) you must include that term (0x² in this case)

Divide. Ex2. Ex3. Ex4.

12 – 6 Adding and Subtracting Rational Expressions To add or subtract rational expressions, you must have a common denominator (just like rational numbers) Often you will have to factor the two denominators to see what you must multiply by to come up with the common denominator Remember that whatever you multiply the denominator by, you must do the same to the numerator

Add or subtract Ex1.Ex2. Ex3.Ex4.

12 – 7 Solving Rational Equations A rational equation contains one or more rational expressions One method for solving rational equations is multiplying through (multiply both sides by something that will eliminate all fractions) Ex1. Solve Remember to always check your answers before you move on by plugging them back in

Ex2. Solve. Open your book to page 673 example 3 When given a rational proportion, cross multiply and solve Keep in mind that you may still have to check for extraneous solutions (b/c the denominator cannot = 0) Solve. Ex3.Ex4.