Chapter 9 Review. By Charlie and Amy

Transcription

1 Chapter 9 Review By Charlie and Amy

2 9.1- Inverse and Joint Variation- Explanation There are 3 basic types of variation: direct, indirect, and joint. Direct: y = kx Inverse: y = (k/x) Joint: y=kxz k is the constant.

3 9.1- Inverse and Joint Variation- Example Example: 1. The variable y varies jointly with x and z. When x=2 and z=5, y=20. Write an equation and solve for y when x = 3 and z = 5. 20=k(2)(5) => 20=10k => k=2 y=2xz => y=(2)(3)(5) The variable y varies inversely with x. When x=7, y=4. Write an equation and solve for y when x=2. 4=(k/7) => k=28 => y=(28/x) => y=28/2= 14

4 Steps to graphing rational functions : 1. Factor (if possible) and find domain 2. Find holes and plot them 3. Find vertical asymptotes, draw them 4. Find y-intercept and x-intercept. Plot both 5. Find horizontal asymptote if there is one and draw 6. Find oblique asymptote if there is one and draw 7. Create an interval table 8. Use the interval table to determine behavior near asymptote 9. Put all together to finish

5 9.2- Graphing Simple Rational Functions- Explanation Rational functions are functions with polynomials on the numerator and denominator An asymptote is a line which the graph would go closer and closer to but not actually touch. Horizontal Asymptotes When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. When the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the coefficient of the largest exponent on the numerator divided by the coefficient of the largest exponent on the denominator. In other words, the quotient of the q values from factoring and the p/q rule. Vertical Asymptote To find the vertical asymptote, solve for x when y is zero for the polynomial function in the denominator. The number(s) is what x can t be because the denominator cannot be zero, which is why the number(s) is the vertical asymptote. X- intercept and y-intercept To find the x-intercept, set y as zero and solve for x. To find the y-intercept, set x as zero and solve for y. Domain and Range The domain is all real numbers except the vertical asymptote. The range is all real numbers except the horizontal asymptote.

6 9.2- Graphing Simple Rational Functions- Example 1. Example: y = x/(x+3) Factored Domain: x -3 Vertical Asymptote: x=-3 Horizontal Asymptote: y=1 Intercepts: (0,0)

7 9.3- Graphing Complex Rational Functions- Explanation Holes: When you factor a rational function and there is a factor in both the denominator and numerator, then that is a hole. Plug in the x to find out the y and that is the coordinate for the hole. Oblique asymptotes: Oblique asymptotes are slanted. There could be either a horizontal or oblique asymptote, but not both. When the degree of the numerator is greater than the degree of the denominator by one, then use long polynomial division. The remainder doesn t matter. The oblique asymptote equation is y = whatever the quotient is. Interval table: To make an interval table, the left side is divided into interval, x, and f(x). You would have to have intervals starting from negative infinity to the vertical asymptotes and x-intercept, then to positive infinity. You can choose any number in the interval for x and substitute it to find out y. Plot those points.

8 9.4- Multiplying and Dividing Rational Functions- Explanation To simplify a rational expression, first factor the numerator and denominator, and then you can cross out any factors that are in both the numerator and the denominator. When you divide a rational expression, you just multiply the first expression with the reciprocal of the second. Multiplying rational expressions has the same process as multiplying fractions.

9 9.5-Addition, Subtraction, and Complex Fractions In order to add or subtract rational expressions, you first have to find a common denominator and then you can add/subtract like any other fraction. Plug in a number other than zero or one to check your answer. A complex fraction is a fraction with a fraction in its numerator and denominator. To simplify a complex fraction, solve the the numerator and the denominator. Then multiply the numerator fraction with the reciprocal of the denominator.

10 9.6- Solving Rational Functions Methods: Fraction Busting: When you fraction bust, you multiply both sides of the equation with the least common denominator so there are no more fractions, hence the name fraction busting. Cross-Multiplying: Cross multiplying rational functions is done the same way as normal fractions. Multiply the numerator with the opposite denominator and multiply the denominator with the opposite numerator.

11 FUN AND GAMES ARE NEXT!