Chapter 9 Notes SN AA U2C9

Transcription

1 Chapter 9 Notes SN AA U2C9 Name Period Section 2-3: Direct Variation Section 9-1: Inverse Variation Two variables x and y show direct variation if y = kx for some nonzero constant k. Another kind of variation is called inverse variation if y k x when k 0. k is called the constant of variation. In direct variation, y is said to vary directly with x. In inverse variation, y is said to vary inversely with x. To see whether an equation shows direct or inverse variation, try to get y on a side all by itself. Example 1: xy 10 Example 2: y x 1 Example 3: Given Equation Rewritten Equation Type of Variation x 5 y Example 4: x 5y If variables x and y vary directly, you can write an equation of the form y = kx by simply figuring out what k is and substituting it back in the original equation y = kx. Example 5: The variables x and y vary directly. Use the given values of x = 5 and y = 2 to write an equation relating x and y. Then find y when x = 2. If variables x and y vary inversely, you can write an equation of the form y k x by simply figuring out what k is and substituting it back in the original equation y k x. page 1 SN AA U2C9

2 Example 6: The variables x and y vary inversely. Use the given values of x = 5 and y = 2 to write an equation relating x and y. Then find y when x = 2. Example 7: The variables x and y vary inversely. Use the given values of x = an equation relating x and y. Then find y when x = 2. 2 and y = 6 to write 3 Another type of variation is called joint variation. Joint variation is when some amount varies directly as the product of two or more other amounts. If z = kxy when k 0, z varies jointly with x and y. You can write equations for joint variation simply by finding k and substituting k back into the equation z = kxy. Example 8: The variable z varies jointly with x and y. Use the given values of x = 3, y = 8, and z = 6 to write an equation relating x, y, and z. Then find z when x = 4 and y = 7. Example 9: The variable z varies jointly with x and y. Use the given values of x = 6 5, y = 10 3, and z = 8 to write an equation relating x, y, and z. Then find z when x = 4 and y = 7. You can also combine variation types. NOTE: If y varies with x inversely, y is on a side all by itself and x is in the denominator under k. If y varies with x directly, y is on a side all by itself and x is in the numerator with k. If y varies jointly with x and z, y is on a side all by itself and x and z are in the numerator with k. page 2 SN AA U2C9

3 Example 10: z varies directly with y and inversely with x. Example 11: x varies inversely with y and directly with z. Example 12: w varies inversely with x and jointly with y and z. Example 13: The work W (in Joules) done when lifting an object varies jointly with the mass m (in kilograms) of the object and the height h (in meters) that the object is lifted. The work done when a 120 kilogram object is lifted 1.8 meters is Joules. Write an equation that relates W, m, and h. How much work is done when lifting a 100 kilogram object 1.5 meters? Section 9-2: The Reciprocal Function Family Many moons ago, you were taught about rational numbers. Rational numbers, as the name applies, deals with ratios, which are just fractions. Rational functions are functions that have the form of f (x) p(x) where p(x) and q(x) are polynomials and q(x) 0. In this section, p(x) and q(x) will be q(x) polynomials that are linear (in other words, no squares or cubes or things of that sort). If we graph a simple rational function like y 1, which is a reciprocal function since it models inverse x variation, the graph of a rational function such as this is called a hyperbola. You can graph it by finding points that make the equation true and then plotting them. page 3 SN AA U2C9

4 There are a few things you should notice here: The x-axis is a horizontal asymptote. This is because, no matter what value you substitute for x, y will never be zero; it will only approach zero. The y-axis is a vertical asymptote. This is because the y-axis is where x = 0, and substituting 0 for x will cause y to be undefined (so nothing can be graphed there). The domain and range are all nonzero real numbers. The graph has two symmetrical parts called branches. For each point (x, y) on one branch, there is a corresponding point ( x, y) on the other branch. To generalize this for other kinds of rational functions, all rational functions are of the following form: y a x h k. h moves the normal graph of y 1 x left and right and k moves the normal graph of y 1 x up and down. If you graph the hyperbola, the following is true: The graph will have a vertical asymptote at x = h, meaning the domain is all real numbers except for h. The graph will have a horizontal asymptote at y = k, meaning the range is all real numbers except for k. Example 1: Identify the horizontal and vertical asymptotes of the graph of the function. Then state the domain and range. y 3 x 2: y 4 x 6 19 : page 4 SN AA U2C9

5 Example 2: Graph the function y 4 8. State the domain and range. x 5 Left side of asymptote Right side of asymptote x y x y Section 9-4: Rational Expressions In Section 9-2, you were introduced to rational expressions and functions as well as how to graph them. However, the rational expressions with which you have been working have all been in simplified form. To be in simplified form, a rational expression cannot have common factors (other than 1 or 1) in its numerator and denominator. In other words, if you have the same factors in the numerator as the denominator, they will cancel out. Some examples of this are as follows: ac bc a b x 3 5x x(x2 5) x2 5 x 2 x x x x 2 x 1 x 2 x 3 x 1 x 3 Notice that you cannot cancel out factors until they actually ARE factors. Factors are numbers that are ONLY multiplied on each other, not added or subtracted to each other. In the last example above (x + 2), (x 1), and (x + 3) are factors because they are being multiplied on other factors. Also notice that, as in the second example above, sometimes you need to factor out to cancel factors out. Had you not done so, nothing would have been able to be divided out. Therefore, simplifying a rational expression generally requires two steps: Factor the numerator and denominator. Divide out any factors that are common to both the numerator and denominator. page 5 SN AA U2C9

6 Example 1: If possible, simplify the rational expression. 3x 2 3x 6 : x 2 4 x 3 27 x 3 3x 2 9x : #3: 15x 2 8x 18 20x 2 14x 12 : Multiplying entails the same principles: multiply the numerators and denominators, respectively, and simplify as above. Also, if you see something in the numerator that can be cancelled out in the denominator, do it. One more thing: note which numbers cannot be substituted because they would make the original function or final function undefined. These are referred to as restrictions. Example 2: Multiply the rational expressions. Simplify the result and state any restrictions. 80x 4 y 3 xy 5x 2 : page 6 SN AA U2C9

7 x 3 2x 8 6x2 96 x 2 9 : #3: x 3 x 3 3x 2 x2 2x 1 : As you have also learned before, dividing is the same thing as multiplying by the reciprocal. For example, x 3 x 2 x 1 x 3 x 2 1 x 1 and x 3 x 2 1 x 1 x 3 x 2 x 1 1 Example 3: Divide the rational expressions. Simplify the result and state any restrictions. 2xyz x 2 z 2 6y 3 3xz : x 2 6x 27 3x2 27x : x 5 page 7 SN AA U2C9

8 Example 4: Almost all of the energy generated by a long-distance runner is released in the form of heat. The rate of heat generation h g and the rate of heat released h r for a runner of height H can be modeled by h g = k 1 H 3 V 2 and h r = k 2 H 2 Write the ratio of heat generated to heat released. When the ratio of heat generated to heat released equals 1, how is height related to velocity? Does this mean that a taller or a shorter runner has an advantage? Section 9.5: Adding and Subtracting Rational Expressions When you first learned to add and subtract fractions, you learned that you need a common denominator to perform addition or subtraction of fractions. The same applies when adding and subtracting rational expressions. First try adding and subtracting rational expressions with common denominators and simplify them. Example 1: Perform the indicated operation and simplify x x 2 10x : 2 5x 2 x 8 5x x 8 : #3: x x 2 5x 5 x 2 5x : page 8 SN AA U2C9

9 To add and subtract rational expressions, it is just like adding and subtracting normal fractions: find a least common denominator, rewrite each fraction with that common denominator, and perform the indicated operation. To get some practice, try finding the least common denominator of the following rational expressions. Recall that, to do so, it helps to factor the denominators first and see what is already in common. Example 2: Find the least common denominator. 4 21x 2, x 3x 2 15x : 1 x 2 3x 28, x x 2 6x 8 : Now it is time to actually start adding and subtracting rational expressions. Example 3: Perform the indicated operation(s) and simplify. 4 7x 5 3x : 2 5x x x 2 : page 9 SN AA U2C9

10 #3: 10 3x x 1 5 6x : As infuriating as it is to work, there are types of fractions called complex fractions that contains a fraction or fractions in the numerator and/or denominator. Such fractions can be simplified using the same techniques that you have just used. Example 4: Simplify the complex fraction. 20 x : x 1 1 x x x 1 1 : 3 x page 10 SN AA U2C9

11 #3: 1 4x 3 5 3(4x 3) : x 4x 3 Section 9-6: Solving Rational Equations Solving rational equations is much like solving equations you have solved before. Most people do not like solving equations with fractions, though, so getting rid of the denominators of the equation help. To do so, remember a few helpful hints: Multiply each term on both sides of the equation by the least common denominator. Simplify and solve the resulting polynomial equation. Double-check for extraneous solutions. Example 1: Solve the equation by using the LCD. Check each solution. 3x x 1 6 2x 7 x : 7x 1 10x 3 1 : 2x 5 3x page 11 SN AA U2C9

12 The last example would have also worked using cross-multiplying once you simplified so that there was only one fraction on each side of the equation. This helps to avoid even having to think about using a least common denominator. Example 2: Solve the equation by cross-multiplying. Check each solution. 8(x 1) x x 2 : 1 x 3 x 4 x 2 27 : Again, remember the following: Look for least common denominators to simplify or combine. If you can, multiply through by the LCD to solve. Cross-multiply when you can. It saves the headache of having to deal with fractions. ALWAYS look for extraneous solutions by double-checking your work. page 12 SN AA U2C9