Can you. 1. 3xy y. Inverse inverse neither direct direct. 6. x y x y

Transcription

1 Salisbur CP Unit 5 You Can (No Calculator) You should be able to demonstrate the following skills b completing the associated problems. It is highl suggested that ou read over our notes before attempting this You Can. More practice problems can be found in the homework, other problems in the tetbook, and on the Big Ideas website. Can ou Recognize and solve direct and inverse variation problems, including word problems For #-7, tell whether and show direct variation, inverse variation, or neither = Inverse inverse neither direct direct Direct (ratios are constant) Inverse (products are constant) 8. If varies directl as and =35 when =7, find when =. = k 35 = k(7) k = 5 = 5 = (5)() = 5 9. If varies inversel as and = when =, find when =. = k = 4 = k k = 4 = 4 = 4 0. When temperature is held constant, the volume V of a gas is inversel proportional to the pressure P of the gas on its container. A pressure of 3 pounds per square inch results in a volume of 0 cubic feet. What is the pressure if the volume becomes 0 cubic feet? = k 0 = k 3 k = = 640 = 64 lbs/in

2 . The length S that a spring will stretch varies directl with the weight F that is attached to the spring. If a spring stretches 0 inches with 5 pounds attached, how far will it stretch with 5 pounds attached? = k 0 = k(5) = 4 5 = ( 4 5 )(5) k = 4 5 = inches Determine under what conditions a rational epression is undefined. 3 8 p p p 8 80 Set denominators equal to zero and solve. =, = - 3. (-)(-)=0; = 4. (-8)(-0); = 0, = 8, = 0 Multipl and divide rational epressions ab 5. a b c d ac d 3 6. a b 5 ab a 3 b a b a b c 3 d ab 4 d ac 3 d (+3) ( 4)(+) ( + 3)( + ) (4 3)(4+3) (4+3) (5+)(5 ) (5+) (4 3)(5 ) ( 8)( 3) ( 0)( 5) ( 8)( 0) ( 4)( 5) 3 4

3 Add and subtract rational epressions Get common denominators k k 3 7 6k 9 k k + 3 (k + 3) 7 (k + 3) (k + 3) 5 (k + 3) ( ) ( + ) 3. Given a graph of a rational function, identif characteristics of rational functions including domain and range, asmptotes, intervals of increasing and decreasing, and end behavior, using appropriate mathematical notation. a. f() = 3 b. f() = c. f() = Domain (, )(, ) Range (, 3)( 3, ) Domain (, )(, ) Range (, 3)(3, ) Domain (, )(, ) Range (, 5)(5, ) Equations for the asmptotes VA: = HA: = 3 Equations for the asmptotes VA: = HA: = 3 Equations for the asmptotes VA: = HA: = 5 Intervals of increasing: none Intervals of decreasing: (, )(, ) Intervals of increasing: none Intervals of decreasing: (, )(, ) Intervals of increasing: (, )(, ) Intervals of decreasing: none End behavior as, 3 as, 3 End behavior as, 3 as, 3 End behavior as, 5 as, 5

4 4. Graph a rational function using transformations. Label and show asmptotes with dashed lines. VA: solve for denominator; HA: ratio of leading coefficients of numerator and denominator a. f() = b. f() = 3 c. f() = +4 3 VA: = HA: = 5 VA: = 3 HA: = VA: = /3 HA: = /3 5. Rewrite the function in the form g() = a + k. Graph the function. Describe the graph of g as a transformation h of the graph f() = a. a. f() = 5 +3 a. Use long division to get f() = one unit up, three units left, reflected over -ais b. f() = a. Use long division to get f() = one unit up, three units left, reflected over -ais VA: = 3 HA: = VA: = HA: =. 5

5 6. Solve a rational equation and identif etraneous solutions. Remember to test the solutions into the denominators to make sure the don t result in a zero in the denominator. If the do, the are etraneous solutions and should not be part of the solution set. a. 3 one fraction = one fraction: cross multipl 3() = ( + ) 3 = + = b. 9 t 4 t3 t3 4 more than one fraction on a side, multipl both sides b LCD to get rid of fractions 9 4(t 3) t 3 = 4(t 3) t 4 t 3 + 4(t 3) 4 4(9) = 4(t 4) + (t 3)() c. + + t = = = +4 + (+)( ) factor first to help find LCD: ( )( + ) ( )( + ) + ( )( + ) + = ( )( + ) + 4 ( )( + ) ( ) + ( + ) = ( + 4) = + 4 =, but is etraneous so no solution

6 7. For an application involving rational functions, write an equation representing the situation and solve. For each problem below, write an equation represent the situation and define the variables. Solve the problem. a. Jason can water all the plants at the botanical garden in 3 minutes. Celia can water them in 5 minutes. If the work together, how long will it take for them to water the plants? What portion of the job Jason can do in a minute + What portion of the job Celia can do in a minute = What portion of the job the can both do in a minute working together b. How man liters of 0% alcohol solution should be added to 40 liters of a 50% alcohol solution to make a 30% solution? Goal % = orig% orig amt + new% New Amt Total Amt 0.30 = (0.50)(40) + (0.0)() 40 + = 80, so add 80 liters = = 4.04 so about 4 minutes