MAT 170 Problem Set #1 - Algebra

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Arizona State University *

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Mathematics

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Apr 3, 2024

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docx

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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. (8,0), (-2,0), (0,8/5)
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. They are the same (same asymptotes and intercepts)
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 ) = 2 ( x + 6 )( x − 7 ) ( x + 3 )( x − 4 ) 6) Please describe how you designed h ( x ) to fulfill each of the listed characteristics. Using the given intercepts and asymptotes I was able to create the function that I wrote above (for example, you can see that the denominator contains the vertical asymptotes)
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