108 Section 5.1 Model Inverse and Joint Variation Remember a Direct Variation Equation y k has a y-intercept of (0, 0). Different Types of Variation Relationship Equation a) y varies directly with. y k b) y varies inversely with. k y c) z varies jointly with and y. z ky d) y varies inversely with the square of. k y e) z varies directly with and inversely with y. k z y The nonzero constant k is called the constant of variation. Eample: 1) The variables and y vary inversely, and y 8 when 3. a. Write an equation that relates and y. b. Find y when 4 Eample: ) Do and y show direct variation, inverse variation, or neither? a. y 4.8 b. y 1.5 c. y 3 Eample: 3) Does the following table of data show inverse variation? If so, fins a model. W 4 6 8 10 H 9 4.5 3.5 1.8 Eample: 4) the ideal gas law states that the volume V (in liters) varies directly with the number of molecules n (in moles) and temperature T (in Kelvin) and varies inversely with the pressure P (in kilopascals). The constant of variation is denoted by R and is called the universal gas constant. a. Write an equation for the ideal gas law. b. Estimate the universal gas constant if V 51.6 liters; n 1 mole; T 88 K; P 9.5 kilopascals.
109 Section 5. & 5.3 Graphing Rational Functions Rational functions are quotients of polynomial functions. This means that rational functions can be epressed as: p () f() q () where p() and q() are polynomial functions and q() 0. The domain of a rational function is the set of all real numbers ecept the -values that make the denominator zero. Vertical asymptotes of a rational function: To find a vertical asymptote, set the denominator equal to 0 and solve. These values become your vertical asymptote(s). Caution: If you can factor the numerator and the denominator and cancel a factor, it becomes a point of discontinuity, (hole) and not a vertical asymptote. Parent rational function: Graph 1 y Graph 1 y Asymptotes: Asymptotes: The transformations that we have used all year to graph functions will work with rational functions. a If we look at the rational function y k a is our outside coefficient and will vertically stretch h or compress the equation or if it is negative, it will reflect it over. The denominator h is considered the inside and will move the graph left/right. The k value will move the graph up/down.
110 Eample: Graph y 1 3 y y y Asymptotes: y y y Graph 3 y 1 Asymptotes: Facts to know and use when graphing rational functions in the form: y p ( ) q ( ) 1) The -intercepts of the graph are the solutions of the numerator p. ( ) p ( ) 0 and solve. ) To find the y-intercept, plug 0 in for and simplify. Discontinuities in the graph: 3) Vertical asymptotes are equations of vertical lines where the denominator is zero. c 4) Holes are found using the binomials that are factored and canceled when simplifying the rational function. Set the binomial equal to zero and solve to get the -value. To find the y- value of the hole, plug the -value into the simplified rational function. Holes are always ordered pairs!
5) Horizontal Asymptote of a rational function: There are three scenarios to look for when finding horizontal asymptotes. a) If the degree of the numerator is less than the degree of the denominator, then automatically y = 0 (the -ais) is the horizontal asymptote. b) If the degree of the numerator equals the degree of the denominator, then keep the two matching degree terms and simplify them to y = leading coefficient of numerator leading coefficient of denominator. c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes. SPECIAL CASE: When the numerator is eactly one degree more than the denominator, they are called slant asymptotes and can be found using long division (disregarding the remainder). It will be in the form of y m b. Graphing Rational Functions by Hand Let p () f() define a rational function in lowest terms. To sketch its graph, follow these steps. q () STEP 1: Find all vertical asymptotes. STEP : Find all horizontal asymptotes or slant asymptotes. STEP 3: Find the y-intercept, if possible, by evaluating f (0). 111 STEP 4: Find the -intercepts, if any, by solving f() = 0. (These will be the zeros of the numerator p().) STEP 5: Determine whether the graph will intersect its non-vertical asymptote y = b by setting the rational equation equal to the horizontal asymptote value and solve for. If is a real number, the graph crosses the asymptote at this -value. STEP 6: Plot selected points as necessary. Choose an -value in each interval of the domain determined by the vertical asymptotes and -intercepts. STEP 7: Complete the sketch. Graph: 3 f( ) 4 Equation(s) of Vertical Asymptotes: Equation(s) of Horizontal Asymptotes: Equation(s) of Slant Asymptotes: -intercepts: y-intercepts: hole:
11 Graph: 3 f( ) ( )( 5) Equation(s) of Vertical Asymptotes: Equation(s) of Horizontal Asymptotes: Equation(s) of Slant Asymptotes: -intercepts: y-intercepts: hole: Graph: f( ) 1 Equation(s) of Vertical Asymptotes: Equation(s) of Horizontal Asymptotes: Equation(s) of Slant Asymptotes: -intercepts: y-intercepts: hole: 1 Graph: f( ) Equation(s) of Vertical Asymptotes: Equation(s) of Horizontal Asymptotes: Equation(s) of Slant Asymptotes: -intercepts: y-intercepts: hole: Graph: f( ) 3 5
113 Equation(s) of Vertical Asymptotes: Equation(s) of Horizontal Asymptotes: Equation(s) of Slant Asymptotes: -intercepts: y-intercepts: hole: 4 Graph: f( ) 1 Equation(s) of Vertical Asymptotes: Equation(s) of Horizontal Asymptotes: Equation(s) of Slant Asymptotes: -intercepts: y-intercepts: hole: Section 5.4 Multiply and Divide Rational Epressions. A Rational epression is in simplified form provided its numerator and denominator have no common factors (other than 1 ). To simplify a rational epression, apply the following property: Let a, b, and c be nonzero real numbers or variable epressions. Then ac a bc b Simplifying a rational epression usually requires steps. Eample: 5 1. Factor the numerator and denominator.. Divide out common factors to both the numerator and the denominator. Eample: 1) Simplify the ration epression. 56 1
114 Eample: ) Multiplying monomial rational epressions. 6 y 10 y 3 3 4 y 3 Eample: 3) Multiplying polynomial rational epressions. 3 3 7 3 4 1 3 1 3 Eample: 4) Multiplying rational epressions by a polynomial. 3 7 8 9 64 To divide one rational epression by another, multiply the first epression by the reciprocal of the a c a d ad second epression. then simplify ad if possible. b d b c bc bc Eample: 5) Dividing polynomial rational epressions. 3 3 4 8 6
115 Eample: 6) Dividing polynomial rational epressions. 8 103 (4 ) 4 Eample: 7) Multiplying and dividing polynomial rational epressions. ( 3) 4 9 Section 5.5 Add and Subtract Rational Epressions. Like Bases: Basically, you just keep the same denominator and add/subtract the numerators. (BE VERY CAREFUL WHEN SUBTRACTING AND THE SECOND NUMERATOR IS a polynomial.) If this is the case, be sure to distribute the negative sign to all that follow it. Eample: 1) With Like bases. a) 3 7 0 b) 3 6 4 4 4 c 3 3 3 3
116 Unlike Bases: Steps to add/subtract ration epressions with unlike bases. Step 1: Factor all of the denominators. Step : Look for Common Binomials, first. (Take the highest power where appropriate.) Step 3: Look for Variable Coefficients. (Take the highest power where appropriate.) Step 4: Look for Numerical Coefficients. (Take the LCM) Step 5. Multiply the top and bottom of the fractions by the Missing parts of the LCD so that all fractions have the same denominator. Step 6: Add/subtract tops and keep the common denominator. Eample: ) With Unlike bases. 4 3 6 3 a) 3 3 1 1 6 9 9 b) To simplify Comple Rational Epressions, you will need to find the LCD of all fractions contained in the comple fraction. Then use that value to Clear the fractions. From there, just simplify whatever is left.
117 Eample: 3) Simplifying a Comple Fraction. a) 1 b) 3 y y 1 y y 3 You Try: 3 4 1 3 4 1 Section 5.6 Solve Rational Epressions. The trick is to clear the fractions using the least common denominator! Eample: 1) An equation with ONE Solution 3 1 1
118 Eample: ) An equation with an Etraneous Solution 5 5 4 1 1 3 6 1 4 Eample: 3) An equation with TWO Solution Eample: 4) Solving an equation by Cross Multiplying 1 1 Eample: 5) Application The recommended percent p of oygen (by volume) in the air that a diver breathes is given by 660 p d 33, where d is the depth (in feet) of a diver. At what depth is air containing 5% oygen recommended?
119 Etension: Rational Inequalities. Steps to solve a Rational inequality: 1. Get it set to zero. Combine all terms involved as one fraction with a common denominator and simplify. 3. Factor, if necessary. 4. Determine what values will make both the top epression and the bottom epression equal zero.. These will be our critical values 5. Place the critical values on a real number line and use the open or closed circles to represent the inequality being observed. REMEMBER THAT THE VALUES FROM THE DENOMINATOR WILL ALWAYS BE OPEN CIRCLES REGARDLESS OF THE ORIGINAL INEQUALITY. 6. Use the number line with the critical values to set up test intervals using interval notation. 7. Choose a value that falls in the test interval and plug it in to the factored version of your inequality. All you need to determine here is the sign (positive or negative) that would happen once the test value is plugged in. This is called sign analysis. 8. Your answer will be the intervals that satisfy the inequality that is set to zero, if it is < or you are looking for the interval was negative. If it is > or you are looking for the interval was positive. 9. You may be asked to graph the solution and also put it in interval notation. Eample: 1) a. +3 1 b. 4+5 4 +5