MAT 170 Problem Set #1 - Algebra

MAT 170 – Problem Set #1 – Explorations / Learn by Doing Graphs of Rational Functions (Prior Material Review): Using Desmos ( https://www.desmos.com/calculator ), graph the following rational function and then sketch its graph below: f ( x ) = 0.5 x 2 − 3 x − 8 x 2 − 4 x − 5 Add the graphs of dashed vertical lines at x =− 1 and at x = 5 along with a horizontal line at y = 0.5 . 1) What are these dashed lines called? Asymptotes 2) Provide the coordinates for all x -intercepts and the y -intercept and label them on the graph. x intercepts: (-2, 0) (8, 0) y intercept: (0, 1.6) 3) Compare your graph to the rational function g ( x ) = ( 0.5 x − 4 )( x + 2 ) ( x + 1 ) ¿¿ then comment on any similarities and any differences. The only thing is their asymptotes, and the intercepts are different. Both graphs look like the same and in the same idea.

4) produce a rough sketch of a graph of a rational function that has the following characteristics: Vertical Asymptotes at x =− 3 and x = 4 with a Horizontal Asymptote at y = 2 . The rational function also has intercepts of (− 6,0 ) , ( 7,0 ) , and ( 0,7 ) . 5) Create a rational function h ( x ) that has these characteristics h ( x ) = ( x + 3 )( x − 4 ) 2 ( x + 6 )( x − 7 ) 6) Please describe how you designed h ( x ) to fulfill each of the listed characteristics. Find g(x) using the vertical asymptotes so it would be g(x) = (x+3)(x-4) Used the horizontal asymptote to determine the degree of f(x) and the leading coefficient ratio. The degree of f(x) is 2 and the leading coefficient ratio of f(x)/g(x) is 2. Use x-intercepts to get f(x). We find that f(x) = (x+6)(x-7) And finally I put everything together and get the above answer. 7) Use Desmos to graph your created function as a final check. Does it fit? Used the Desmos to graph

“Pay it Forward” Exponential Growth Activity (Learn by Doing – Preview of up coming content) The story: Young Trevor, is caught up by an intriguing assignment from his new social studies teacher, Mr. Simonet. The assignment: think of something to change the world and put it into action. Trevor conjures the notion of paying a favor not back, but forward--repaying good deeds not with payback, but with new good deeds done to three new people. Trevor's efforts to make good on his idea bring a revolution not only in the lives of himself, his mother and his teacher, but in those of an ever-widening circle of people completely unknown to him. First you need make some assumptions:  How Long does each person get to complete the 3 good deeds? (1 day, 1 week, 1 month…)  Once someone has a good deed done to them they do not receive another good deed until after everyone else receives one. Now complete the table for at least seven rounds of good deeds and see if you can find an equation to fit the growth process. The equation is an=a1 ⋅ r(n−1) Time in: _________ Number of New People Helped Total People Helped Start (0) 1 1 1 3 4 2 9 13 3 27 40 4 81 121 5 243 364 6 729 1093 7 2187 3280 n

Questions to consider once you have found and tested your mathematical model... 1. How many new people would be helped during the 12 th round of good deeds? What is the total number of people who have been helped by the end of the 12 th round? 53144 and 797160 2. There are about 4.7 million people in the Metro Phoenix area, how long would it take for everyone in the Phoenix Metropolitan area to receive a good deed? (Census Bureau 2017) 14 3. There are about 7.3 million people in Arizona, how long would it take for everyone in Arizona to receive a good deed? (Census Bureau 2019) 15 4. There are about 328.2 million people in the USA, how long would it take for everyone in the USA to receive a good deed? (Census Bureau 2019) 18 5. There are 7.7 billion people in the world, how long before everyone in the world would receive a good deed? 21 6. What do the numbers tell you about the possibility of changing the world for the better? That it is not hard to change it 7. What is needed to make it happen? 3 people need to spread message to be able to help 3 other people