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Graph Systems of Linear Inequalities Peter Paliwoda Multi-step video game problem • An online media store is having a sale, as described in the ad shown. Use the information in the ad to write and graph a system of inequalities for the regular video game prices and possible sale prices. Then use the graph to estimate the range of possible sale prices for games that are regularly priced at $20. ONE DAY SALE! SAVE 30%-70% ON ALL VIDEO GAMES REGULAR PRICE: $20-$50) System of Inequalities • • • • x>=20 x<=50 y>=0.3x y<=0.7x Regular price must be at least $20 Regular price can be at most $50 Sale price is at least (100-70)%=30% of regular price Sale price is at most (100-30)%=70% of regular price ONE DAY SALE! SAVE 30%-70% ON ALL VIDEO GAMES REGULAR PRICE: $20-$50) Graph Inequalities Sale Price (dollars) y x>=20 x<=50 x Regular Price (dollars) Identify the Range of Possible Sale Prices • From the graph you can see that when x=$40, the value of y is between these values: 0.3(40)=12 and 0.7(40)=28 So, the value of y satisfies 12 <= y <= 28 Modeling Bicycle • Speed of a Bicycle: The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4 inches, 2 inches, and 14 inches respectively. A cyclist is pedaling at a rate of 1 revolution per second. Find Speed of Bicycle in ft/min pedal _ sprocket 2 r 2 4 " 25 . 13 " / sec wheel _ sprocket 2 r 2 2 " 12 . 57 " / sec pedal _ sprocket wheel _ sprocket 25 . 13 2 12 . 57 wheel _ of _ bicycle 2 r 2 14 " 88 " 2 _ wheel _ turns _ for _ each _ pedal _ sprocket _ turn speed _ of _ bicycle ( 2 14 2 )" sec 176 " 1 ft 14 . 7 ft sec 12 " sec Write function for Distance d (in miles) in terms of n revolutions of the pedal n 2 4 2 2 2 2 14 ( n 2 2 14 )" 1mile 63360 " n ( 0 . 002776 ) miles Write a function for the distance d (in miles) a cyclist travels in terms of time (in seconds). n 2 4 2 2 2 14 ( n 2 2 14 )" 1mile 63360 " n ( 0 . 002776 ) miles sec n ( 0 . 002776 ) miles sec Why should you learn Inverse functions • It is useful in manipulating graphs and the way they are displayed • Recall that to find the inverse of a line of points f(x)={(1,5),(2,6),(3,7),(4,8)} is done by interchanging the first and second coordinate • f-1(x)={(5,1),(6,2),(7,3),(8,4)} • So, f is equal to the range of f-1 and vice versa. Problem • The factory sales f (in millions of dollars) of digital cameras in the United States from 1998 to 2003 are shown in the table. Year, t Sales, f(t) (in millions) 1998 519 1999 1209 2000 1825 2001 1972 2002 2794 2003 3421 Questions Year, t Sales, f(t) (in millions) 1998 519 1999 1209 2000 1825 2001 1972 2002 2794 2003 3421 • Does f-1 exist? • If f-1 exists, what does it represent in the context of the problem? • If f-1 exists, find f-1 (1825) • If the table was extended to year 2004 and if the factory sales of digital cameras for that year was $2794 million, would f-1 exist? Explain Does f-1 exist? Yes, it does exist! f(x) Year, t Sales, f(t) (in millions) 1998 519 1999 1209 2000 1825 2001 1972 2002 2794 2003 3421 f-1(x) Sales, f(t) (in millions) Year, t 519 1998 1209 1999 1825 2000 1972 2001 2794 2002 3421 2003 If f-1 exists, what does it represent in the context of the problem? • f-1 represents the domain of f in f(x) x f(x) Domain Range in f-1(x) x f-1(x) Domain of f-1(x) Range of f-1(x) Range of f(x) Domain of f(x) If f-1 exists, find f-1 (1825) Year, t Sales, f(t) (in millions) 1998 519 1999 1209 2000 1825 2001 1972 2002 2794 2003 3421 f-1(x) Sales, f(t) (in millions) Year, t 519 1998 1209 1999 1825 2000 1972 2001 2794 2002 3421 2003 If the table was extended to year 2004 and if the factory sales of digital cameras for that year was $2794 million, would f-1 exist? Explain • No, because the extended function would fails the horizontal line test. More than one horizontal intersection