Ch+9+0910

Rational Function Graph

Hello! Welcome to this wiki. It is time to review rational expressions.Please complete the questions below using this wiki page. Place your answers under the discussion tab at the top of this page.

1. What value of may not be used for k with inverse variation?

__2. What variable is varying jointly in the joint variation definition?__ _

__3. What equation is given for direct variation in the definition?__

__4. What is the answer to 9.1 Practice Problem #2?__

5. What is the name of the graph of a rational expression in 9.2? _

6. What is the domain and range for 9.2 Practice Problem #6? _

7. In 9.3, what are the asymptotes and x intercepts of #7? _

__8. In 9.5, what is the answer to #14?__ _

9. In 9.6, what is stated as a real-life example to use solving rational equations? _

__10. What was the video about?

Mary Sweitzer - Notes Cheyenne Stambaugh - Notes Alex Schmitt - Multi-Media Devin Taylor - Real Life Gabriella Sallada- notes, editor Amanda Saravia- Practice Problems__ **9.1 Inverse and Joint Variation** __ Definitions: * __Inverse Variation __– y=k/x ; where k cannot be 0
 * __Constant of variation __– nonzero constant k; y varies inversely with x__

__Real Life- Finding the speed of a whirlpool's current.
 * Joint Variation __– a quantity varies directly as the product of two or more other quantities; z=kxy where z cannot be 0__
 * Direct Variation __– y = kx__

Tell whether x and y are direct variation or inverse variation or neither.__ __ 1. xy = 8 2. x = y/3 The variables x and y vary inversely. Use the given values to write and equation relating x and y. 3. x = 10, y = 2 The variables z varies jointly with x and y. Use the given values to find an equation that relates the variables. 4. x = 4, y = 3, z = 24 __
 * Practice Problems **

**9.2 Graphing Simple Rational Functions** __ Definitions: - x- axis is a horizontal asymptote - y- axis is a vertical asymptote - The domain and range are all nonzero real numbers - For each point (x,y) there is a corresponding point (-x,-y) on the other branch All rational functions of the form y = (a/ x – h) + k will have asymptotes at y = k and x = h. To draw this graph, choose some x values, plug them into the equation and plot them. Finally connect the dots and draw in the asymptotes Real Life- Finding the frequency of an approaching ambulance siren.
 * Rational Function – function of the form f(x) = p(x)/q(x)
 * Hyperbola – graph of a rational function
 * Branches – two symmetrical parts of a hyperbola

Graph the function. State the domain and range. 5. y = -2/x 6. y = 2/(x-3) + 1 __
 * Practice Problems **

__9.3 Real Life- Finding the energy expenditure of a parakeet. State the asymptotes and x intercepts. 7. y = x/x^2 + 4 8. y = x^2 + 2 / x^2 - x - 6 __**9.4 Multiplying and Dividing Rational Expressions** __When simplifying rational expressions, the following property can be used:
 * Practice Problems **

[Let a, b, and c be nonzero real numbers or varible expressions.]

ac/bc= a/b <=== The common factor of c can be divided out

The same rules that apply to multiplying numerical fractions applies when multiplying rational expressions. The numerators are multiplied, along with the denominators, and then just written as a new fraction in simplified form.

a/b x c/d ---> a x c / b x d = ac/bd <== this can even be simplified, if possible.

When dividing rational expressions, just take the first expression multiplied by the recipricol of the second expression.

a/b / c/d > a/b x d/c = ad/bc <==Simplify, if possible

Real Life- Finding the average number of acres per farm. Simplify the rational expression, if possible. 9. y^2 - 81 / 2y - 18 10. x + 3 / x^2 + 6x + 9 Multiply the rational expressions and then simplify. 11. x^2 + 2x - 3 / x + 2 times x^2 + 2x / x^2 -1 12. 12 - x / 3 times 3 / x - 12 Divide the rational expressions and then simplify. 13. 48x^2 / y divided by 36xy^2 / 5 14. x^2 / x^2 - 1 divided by 3x / x + 1 __
 * Practice Problems **


 * 9.5 Addition, Subtraction and Complex Fractions**

__ How to solve the addition and subtraction of complex fractions depends on whether or not the expressions have //like// or //unlike// denominators. If they have like denominators then simply add or subtract the numerators and then put the result over the common denominator.

If they have unlike denominators, then you need to find the LCD (Least Common Denominator). To do this, just multiply the denominators of each expression by each other as if they were one. (i.e multiply by x+3/ x+3 instead of just x+3) Then rewrite the expression and simplify.

__Complex Fractions __ Complex fractions are formed of two fractional expressions, one on top of the other. There are two methods for simplifying complex fractions. The first method is fairly obvious: find common denominators for the complex numerator and complex denominator, convert the complex numerator and complex denominator to their respective common denominators, combine everything in the complex numerator and in the complex denominator into single fractions, and then, once you've got one fraction (in the complex numerator) divided by another fraction (in the complex denominator), you flip-n-multiply. (Remember that, when you are dividing by a fraction, you flip the fraction and turn the division into multiplication.) __ __3+ 2/ x^2 (3x^2/x^2 + 2/x^2) (3x^2+2/x^2) = (4x+1/x)(x^2/3x^2+2) <-- an x can be taken out of both fractions to simplify. = x(4x+1) = __ _4x^2+x __3x^2+2 3x^2+2 || Everything cancels at this point, so this is the final answer. 4x^2+x /3x^2+ 2
 * This method looks like this: ||  || 4+ 1/x_ __=__ (4x/x + 1/x) __=__ (4x+1/x)
 * Multiply*__

Real Life- Modeling the total number of male college graduates. **Practice Problems** Find the LCD. 15. 3 / x + 4 x / x^2 - 16 x + 2 / 4 16. 13 / x^2 - 2x + 1 4 / x^2 - 1 5 / x (x + 1) Perform the indicated operations and then simplify. 17. 2x - 1 / x^2 - x - 2 minus 1 / x - 2 18. 2x / x + 2 minus 8 / x^2 + 2x plus 3 / x Simplify the complex fraction. 19. (1 / x + 9 plus 1 / 5) / ( 2 / x^2 + 10x + 9)

**9.6 Solving Rational Equations**__

To solve a rational equation, multiply each term on both sides of the equation by the LCD of the terms. Simplify and solve the resulting polynomial equation.

To solve a rational equation for which each side of the equation is a single rational expression, use **cross multiplying**. When adding and subtracting rational expressions, you had to find a common denominator. Now that you have equations, you are allowed to multiply through (because you have two sides to multiply on) and get rid of the denominators entirely. In other words, you still need to find the common denominator, but you don't necessarily need to use it in the same way.

Example: Least Common Denominator is 3x.
 * Solve the following equation: **

7/x - 1/3x = 5/3 -> Write the original equation. 3x (7/x - 1/3x) = 3x(5/3) --> Multiply each side by the LCD 21 - 1 = 5x -> Simplify 20 = 5x ->Subtract 4=x --> Divide each side by 5

Solve by using LCD or cross multiplying. Check each solution. 20. 3x/x-2 = 1 + 6/x-2 21. 6+5x/3x = 7/x 22. 6/x - 7x/5 = x/10
 * Practice Problems **

media type="file" key="My First Project.3gp" width="300" height="300" 9.6 Real Life- Finding the year which a certain amount of rodeo prize money was earned.

Answers to Practice Problems 1. inverse 2. direct 3. y = 20/x 4. z = 2xy 5. curve in the 2nd quadrant with x = 0 and y = 0 as asymptotes, another curve in 4th quadrant with x = 0 and y = 0 as asymptotes Domain- x= all real but 0 Range- y = all real but 0 6. curve in first quadrant and 3rd going into fourth quadrant with x = 3 and y = 1 as an asymptotes Domain- x = all real but 3 Range- y = all real but 1 7. Asymptotes- x = 2 and x = -2 and y = 0 X-Intercepts - (0,0) 8. Asymptotes- x = 3 and x = -2 and y = 1 X-Intercepts- no intercepts 9. y + 9 / 2 10. 1 / x + 3 11. x (x + 3) / x + 1 12. -1 13. 20x / 3y^3 14. x / 3 (x - 1) 15. 4 (x + 4) ( x - 4) 16. x (x + 1) (x - 1) (x - 1) 17. 1 / x + 1 18. 2x - 1 / x 19.( x + 14) (x + 1) / 10 20. no solution 21. x = 3 22. x = 2 and x = -2