quiz2-8.40-soln. - y&& 2 the horizantal asymptote...

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MATH 101-B Fall 2004 Quiz 2 ( 8:40-10:30 Group) Oct. 08, 2004 Time: 15 minutes Name: Student No: Follow the directions. No work No credit!! Problem 1 (7 pts.) The graph of y f x ± is shown in the figure. Sketch the graph of y 2 f x /2 ± ± 1. Label any intercepts or asymptotes that can be determined. 5 4 3 2 1 0 -1 -2 -3 -4 -5 2.5 2 1.5 1 0.5 0 -0.5 -1 x y f x ± Solution. The following sequence of changes need to be made on the above graph: Stretch x -axis by a factor of 2; stretch vertically by a factor of 2; flip around the x -axis; shift everything up by 1. In the original graph, y 2 is the y -intercept and y 1 is a horizantal asymptote. The first step doubles x -intercepts, which are in the intervals 2, 1 ± and 1,2 ± . The second step makes y 4 the y -intercept and
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Unformatted text preview: y & & 2 the horizantal asymptote. Third step changes the signs of y-intercept and horizantal asymptote. The last step makes y & & 3 the y-intercept and y & 3 the horizantal asymptote. It also affects the x-intercepts. The resulting graph is: 6 4 2-2-4-6 3 2 1-1-2-3 x y & 2 f & x /2 ± ± 1 Problem 2 (3 pts.) Prove the identity: log a & AB ± & log a & A ± ± log a & B ± Solution. Let’s say log a & A ± & x , log a & B ± & y , log a & AB ± & z . We want to show that x ± y & z . Rewrite logs as exponentials: a x & A , a y & B , a z & AB . Then we have a x a y & a x ± y & a z . Hence, x ± y & z and we are done....
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quiz2-8.40-soln. - y&& 2 the horizantal asymptote...

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