EXPLORING RATIONAL FUNCTIONS GRAPHICALLY

Transcription

1 EXPLORING RATIONAL FUNCTIONS GRAPHICALLY Precalculus Project Objectives: To find patterns in the graphs of rational functions. To construct a rational function using its properties. Required Information: A rational function is a function of the form f g, g s 0 where f() and g() are polynomials. The graphs of rational functions may have horizontal asymptotes, vertical asymptotes, slant asymptotes or "holes". Just as patterns can be found in families of polynomial functions that are useful in sketching the functions, patterns can be found in families of rational functions. The horizontal asymptotes are found by observing the end behavior i.e. behavior as increases or decreases without bounds. The horizontal asymptotes eist whenever the degree of the numerator is less than or equal to the degree of denominator. The vertical asymptotes are the zeroes of the polynomial in the denominator. Slant asymptotes eist when the degree of the numerator is one more than the degree of the denominator and found by computing the quotient.. A "hole" in any function is created at a point when the limit of the function eists at that point but is not equal to the value of the function. Solved Eample : Using a graphing utility (calculator or graphing program), sketch the following rational functions on one set of aes. Discuss their similarities and their differences. Include asymptotes, intercepts, domain and range. f = g = h = 6 Solution: restart:with(plots): Warning, the name changecoords has been redefined f:= /^: g:= /^: h:= /^6: plot([f,g,h], = -.., y = -.., title = "graphs of /^n, n =,, and 6", legend = ["/^", "/^", "/^6"]);

2 graphs of /^n, n =,, and 6 y 0 6 K K3 K K K 3 /^ /^ /^6 The domain of all three curves is the set of all real numbers ecept = 0. The range of all three curves is y 0. = 0 is the vertical asymptote for all three curves. y = 0 is the horizontal asymptote for all three curves. All three curves are in the first and second quadrants. As the eponent increases, the graph is further from the y-ais for - < < and closer to the -ais for < - and. All graphs pass through the points (-, ) and (,). Solved Eample : Define and sketch a rational function which has vertical asymptotes at = - and and a slant asymptote. Solution: A rational function has vertical asymptotes at = - and = if the denominator has factors ( + ) and ( - ). It has a slant asymptote if the degree of the numerator is one more than the degree of the denominator and is obtained by finding the quotient. 3 We define: f() = C K f:= -^3/((+)*( - )); f := / quo(^3,(+)*( - ),); 3 C K K C implicitplot({y=f(), = -, =, y = - +}, = -0..0, y = -0..0); (6.) (6.)

3 0 y 5 K8 K6 K K K5 K0 Solved Eample 3: Write equations for two different rational functions with the given properties:. The horizontal asymptote is y =.. The vertical asymptote is = There is a hole (discontinuity) at =. Sketch the function. Solution: From the solved eample above the horizontal asymptote is y = if the function has a limit as increases or decreases without bounds. If a function is undefined at = -3 then the rational function has a vertical asymptote at that point. The hole in any function is created when the limit eists at = but is not equal to the value of the function. Many functions can be defined to satisfy these conditions. We define two in the following. f:= - piecewise(=,, /( + 3)); f := /piecewise =,, C 3 (8.) implicitplot({y=f(), =-3, y = }, = -0..0, y= -..);

4 y 3 Note how the function, horizontal and vertical asymptotes are plotted. Maple does not do a good job of plotting the rational function near = -3. g:= - piecewise(=,, ^/( + 3)^); K0 K5 K 5 0 K K3 K g := /piecewise =,, C 3 implicitplot({y=g(), =-3, y = }, = -0..0, y= -5..5); (8.) y K0 K K K These two functions satisfy the given conditions. Maple does not plot the point (, ). The functions have a hole discontinuity at =, because lim f = /, while f() = and lim g = /6 while g() =. This can be seen as / / follows: limit (f(), = ); f();limit(g(), = ); g(); 6 (8.3)

5 ASSIGNMENT Problem : Using a graphing utility (calculator or graphing program), sketch the following rational functions on one set of aes. Discuss their similarities and their differences. Include asymptotes, intercepts, domain, and range. f = g = 3 h = 5 Problem : Compare and contrast the graphs of the functions in the solved eample (even powers) and the graphs of the functions in Problem (odd powers). Problem 3: Using a graphing utility (calculator or graphing program), sketch the following rational functions on one set of aes. Discuss their similarities and their differences. Include asymptotes, holes, intercepts, domain, and range. C f = g = K 5 K 3 K 0 Problem : Find all vertical and horizontal asymptotes for each of the given functions. f = 3 g = K K 5 C 6 Problem 5: Find a rational function with vertical asymptotes at = - and = and a slant asymptote. Sketch the function. Problem 6: Write equations for two different rational functions with the given properties:. The horizontal asymptote is y = 5. The vertical asymptotes occur at = - and =. Sketch the function.

6 MSIP Grant #P0A0065: "Three Urban Calculus Reform Programs: Adopting the Best" 00-03