Mathematics 00a Summary Notes page 57 4. Curve Sketching Asymptotes are additional pieces of information essential for curve sketching. Vertical Asymptotes The line a is a vertical asymptote of the graph of a function y f() if either, f() ± or + f() ±. a a Note that ± is not a number but rather a symbolic convenience designed to indicate a direction of indefinite increase or decrease along the real line. A polynomial function has no vertical or horizontal asymptotes. P ( ) A rational function f( ) where P and Q are polynomial functions has a Q ( ) vertical asymptote at a if Q(a) 0 but P(a) 0. Eample. ( ) ( )( + ) ( ) ( + ) ( ) The line is therefore a vertical asymptote of the graph of f( ). ( ) Horizontal Asymptotes The line y b is a horizontal asymptote of the graph of a function y f() if either f() b or f() b. Eample. + + + The line y is therefore a horizontal asymptote of the graph of f( ). + In the case of horizontal asymptotes, it s important to determine whether the asymptote is being approached from above or below. This can easily be accomplished by eamining the value of the function at a convenient large number (in magnitude) compared to the value of the asymptote.
Mathematics 00a Summary Notes page 58 Eamine the function + +. When is million, + the numerator is slightly smaller than while the denominator is slightly larger than. The ratio is smaller than. Therefore, the function approaches the asymptote from below. Suppose in eample. Then / is much smaller than / in magnitude and can be ignored. The numerator is (for all practical purposes) and the denominator is smaller than for very large negative. The ratio will eceed and the asymptote is approached from above. The graphs of functions can also have oblique asymptotes (straight lines not parallel to the principal aes). The function f() + is a case in point. As ±, f() behaves like the line y (the oblique asymptote to its graph). Eample. Let f() +.. Clearly the graph of f has a vertical asymptote, because 0 at while + 0 at.. + graph of f. +. So y is a horizontal asymptote of the. ( + ) ( ) the left. because the ratio looks like + approaching from 4. ( + ) + ( ) the right.. because the ratio looks like + + + approaching from
Mathematics 00a Summary Notes page 59 5. As, + looks like < > <. So the asymptote is approached from below. 6. As, + looks like > < >. So the asymptote is approached from above. 7. Sketch y
Mathematics 00a Summary Notes page 60 An Algorithm for Curve Sketching The last four sections were designed to build a respectable set of tools to facilitate the analysis of curves and curve sketching. These tools together with a few basic properties of the curves themselves are summarized here and used to graph functions in the eamples that follow. An Algorithm for Sketching the Graph of y f(). Determine the domain of f.. Find the and y intercepts of f.. Determine the behavior of f for large absolute values of. 4. Find all horizontal and vertical asymptotes of f. 5. Determine the intervals where f is increasing and where f is decreasing. 6. Find the relative etrema of f. 7. Determine the concavity of f. 8. Find the inflection points of f. 9. Plot a few additional points to help further identify the shape of the graph of f and sketch the graph. Eample 4. Sketch the graph of the function y f() +. solution:. First simplify the improper rational function. 4 y + + y f() is continuous for all. That is D: (, ) (, ).. No intercept because y intercept. + 0 has no real roots.. f as and f as f is unbounded for large absolute.
Mathematics 00a Summary Notes page 6 4. The graph of f has a vertical asymptote at and the oblique asymptote y +. 4 5. f ( ) 0 ± 4 or 0. (0, ) and (4, 7) are critical points. f > 0 f < 0 f < 0 f > 0 0 4 Note that is a split number even though it is not a critical number, the reason being, that the function may change direction across a vertical asymptote. f is increasing on the intervals (, 0) and (4, ). f is deceasing on the intervals (0, ) and (, 4). 6. From the chart and the first derivative test, f has a relative maimum at the point (0, ) and a relative minimum at the point (4, 7). 8 7. f ( ) ( ) f < 0 f > 0 f is concave down on (, ) and concave up on (, ). Note that is a split point even though it does not belong to the domain of f. Concavity may change across a vertical asymptote as is the case here. 8. There are no points of inflection. 9. It is useful to plot the points where f has relative etrema and the asymptotes.
Mathematics 00a Summary Notes page 6 y 0 8 y + 6 4 0-0 -5-0 5 0-4 -6-8 -0 Eample 5. Sketch the graph of the function y f(). solution:. f is defined for all ±. That is D: (, ) (, ) (, ).. Both and y ntercepts at the point (0, 0).. f 0 as and f 0 as f is bounded for large absolute. 4. Vertical asymptotes: ± ; horizontal asymptote: y 0 5. f + ( ) < 0 for ± f has no critical points (f is undefined at ± ) f is decreasing everywhere. 6. f has no relative etrema.
Mathematics 00a Summary Notes page 6 7. ( + ) ( ) f ; f 0 at 0 and undefined at ± f < 0 f > 0 f < 0 f > 0 0 f is concave down on (, ) (0, ) and concave up on (, 0) (, ); 8. (0, 0) is a point of inflection. Note: and are not in the domain of f. 9. f() f() symmetric about (0, 0) 5 4 y 0-5 - - - 5 - - -4-5 Eample 6. Sketch the graph of the function y f(). + ( ). Factoring here is useful: y. ( + ) f has domain: (, ) (, 0) (0, ). The y intercepts are y ±. There is no intercept.. f as and f as f is bounded for large absolute. 4. Vertical asymptotes: and 0; horizontal asymptote: y.
Mathematics 00a Summary Notes page 64 5. 4 ( + ) ( )(+ ) + + f ( ) ( + ) ( + ) + + 0 has no solution so f has no critical points. + + Furthermore, > 0 for 0,. ( + ) So f is increasing everywhere. 6. f has no etreme values. 7. f ( ) (4+ )( + ) ( + + )()( + )(+ ) ( + ) + + + + + + ( + ) ( + ) ( ) 4( )( ) f < 0 f > 0 f < 0 f > 0 / 0 The graph of f is concave up on (, ) (/, 0); and concave down on (, /) (0, ). 8. The graph of f has a point of inflection at (/, ). 9. 0 5 0 5 y -4-4 -5-0 -5