What is the reasonable domain of this volume function? (c) Can there exist a volume of 0? (d) Estimate a maximum volume for the open box.

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1 MA Lesson 11 Summer 016 E 1: From a rectangular piece of cardboard having dimensions 0 inches by 0 inches, an open bo is to be made by cutting out identical squares of area from each corner and, turning up the sides and taping (a) Find all positive values of such that V() > 0, and sketch the graph of V for > 0. V() = (0 )(0 ) Hint: Find the zeros and make a sign chart. V V (b) What is the reasonable domain of this volume function? (c) Can there eist a volume of 0? (d) Estimate a maimum volume for the open bo. 1

2 MA Lesson 11 Summer 016 Asymptotes and Holes Consider the following function: 1) y Let or What is the value of y? 9 ( )( ) As gets very, very close to or, the denominator becomes a very, very small decimal. The value of or very, very small decimal The domain of the function above does not include or. D = (, ) (,) (, ). In fact, from the argument above, at or, the function is approaching or -. The graph of the function would be getting greater and greater or lesser and lesser. The graph would be going toward infinity or negative infinity. This indicates there would be vertical asymptotes at = and =. An asymptote is a line that the graph of a function will approach, but probably not intersect. **There is an eception. Some graphs of functions may intersect horizontal asymptotes. However, a graph of a function will never, never intersect vertical asymptotes. E : Usually vertical asymptotes of rational functions may be found by determining what values of made zero denominators. 9 As, y or g ( ) ( )( 5) As 5, y or ( )( 5) 0 0 or Above are the equations for the vertical asymptotes of g. Below is a graph of function g with the asymptotes shown. y

3 MA Lesson 11 Summer 016 E : As seen in eample, usually vertical asymptotes may be found by setting the denominator of a rational function equal to zero. However, if there is a factor of the denominator that cancels with a factor in the numerator, that factor does not yield a vertical asymptote; it yields the location of a hole in the graph. ( ) f( ) 4 ( )( ) The factor cancels from the denominator and numerator. At =, there is a hole in the graph. As, y or. However, as, y a hole in the graph at (, ) The graph of f is the same as the graph of g ( ), ecept for a hole when = or at the point (, 1 ). 1 [ g( ) g() ] There is a vertical asymptote of = and a hole at (, 1 ). 1 simplified f( ) 1 when, f() 4 To determine when a denominator indicates vertical asymptotes or holes (or neither), let s eamine the following (1 ). (1) y 5 As gets very, very large ( ), in the denominator the 5 does not add very much onto the. Basically, we are comparing. As, the denominator is growing or increasing much faster than the numerator,. The result is a value approaching zero. As, y 0 As, y 0 () y 1 As becomes very large, ; the denominator and numerator are growing or increasing at about the same rate. In the numerator, subtracting does not change the much. In the denominator, subtracting the 1 does not change the much. Therefore the resulting ratio, resulting value of. As, y () As, y y faster than the denominator. The resulting ratio is As, y or As, y or As gets very, very large, ; the numerator is growing or increasing much, which will approach infinity.

4 MA Lesson 11 Summer 016 The logical arguments on the previous page lead to the following conclusions about how to find horizontal asymptotes equations of rational functions. Compare the ratio of the term of the highest degree in the numerator to the term of the highest degree in the n a denominator. (Let s let this ratio be.) The horizontal asymptote, if it eists, is given by: k b (1) y 0 ( -ais) if n k In other words, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the -ais. a () y if n k In other words, if the degree of the numerator equals the degree of the b a denominator, the horizontal asymptote is the line y (the ratio of the leading terms in b numerator and denominator). () There is no horizontal asymptote, if the degree of the numerator is greater than the degree of the denominator. As gets very large or very small, the function values get very large or very small as well. One remark about horizontal asymptotes: Vertical asymptotes are sacred ; they will never intersect with the graph of the function. Horizontal asymptotes are not sacred ; they may intersect with the graph of the function, usually close to the origin. E 4: Use asymptotes to sketch the graph of f( ). You might also need a few points and determine if the graph will intersect the horizontal asymptote, if it eists. As sign chart might also help determine when the function is positive and when it is negative. 4

5 MA Lesson 11 Summer 016 E 5: Use asymptotes and a few points to sketch the graph of g ( ). You might also make a sign 6 chart to determine when the graph is above or below the -ais. You might need to determine if the graph intersects the horizontal asymptote and where. 5

6 MA Lesson 11 Summer E 6: Use asymptotes, a few points, and a sign chart to help sketch the graph of f( ) ( ) graph intersect the horizontal asymptote?. Does the 6

7 MA Lesson 11 Summer 016 E 7: Use asymptotes, a few points, and a sign chart to help sketch the graph of h ( ). Does the graph 1 intersect the horizontal asymptote? 7

8 MA Lesson 11 Summer 016 E 8: Use asymptotes, a few point, and a sign chart to help sketch the graph of graph intersect the horizontal asymptote? If so, where? y 6. Does the 8

9 MA Lesson 11 Summer 016 E 9: Use asymptotes, a few points, and a sign chart to help sketch the graph of graph intersect the horizontal asymptote? f( ) 6. Does the 9

10 MA Lesson 11 Summer 016 E 10: Use asymptotes, a few points, and a sign chart to help sketch the graph of g ( ). 10

11 MA Lesson 11 Summer E 11: Use all of the information asked for in previous eamples to sketch the graph of y. Does the 1 graph intersect a horizontal asymptote? 11

12 MA Lesson 11 Summer 016 E 1: Use all of the information you can find to sketch the graph of intersect a horizontal asymptote? f( ) 4 4. Does the graph 1

13 MA Lesson 11 Summer 016 E 1: Write an equation for a rational function f that satisfies the given conditions. Vertical asymptote: = Horizontal asymptote: y = 0 y-intercept: f(0) = 4 E 14: Write an equation for a rational function g that satisfies the given conditions. Vertical asymptotes: =, = Horizontal asymptote: y = -intercepts: (,0), (,0) 1