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0+ 2 x→a means x approaches a from the left x→a* means x approaches a from the right f(x) →0 as X→∞ f(x) 0.1 0.01 0.00001 X 440 Approaching 0 Range -470 (-∞, 0) oco, ∞0) Horizontal Asymptoteo (x-axis) x goes to negative infinity; that is, x decreases without bound 2 x→∞ means x goes to infinity; that is, x increases without bound Asymptotes: Informally speaking, an asymptote of a function is a line to which the graph of the function gets closer and closer as one travels along that line 1. The line x = a is a vertical asymptote of the function y=f(x) if __y approaches to as x approaches a from the right or left HH y →→∞ as x→a yas x→a b y b as x→ ∞0 2. The line y = b is a horizontal asymptote of the function y=f(x) if_ as x approaches ±00 *A rational function has vertical asymptotes where the denominator is zero H H a II. Transformations of y = 1/ y →→ as xa YA y →b as x→→-00 X y →→∞ as x→a VA x = 0 because 1/x 18 undefined y approaches b the function is undefined, that is, where 3 A rational function of the form reflecting r(x) ax+b cx +d can be graphed by shifting, stretching, and/or the graph of f(x) = 1/x using linear fractional transformations 4 Example 2: Using transformations to graph each rational function, and state the domain and range. 2 -7²-3 r(x) = a) Let f(x) = x, then we can expressin terms of fas r(x) = 2(x-3) X-3=0 VA x = 3 Domain (-∞0, 8) 0 (3,00) (ocoدمرم Range 3x +5 X +2 b) Use long division to simplify s(x) s(x) = -23 5 = 3+1/x+2 3 -1 __f(x) = 1/x Bhift left 2 units reflect over x-axis shift up a Domain -2 (-∞03-2) C (-2,00) Range 3 (-∞, 3) (3.00) 5 s(x) = 3x + 5 x + 2 Vertical asymptote x=3 r(x)=√x ²3 Vertical asymptote x = -2 0 Horizontal asymptote y=0] Horizontal asymptote y=3 k 6 III. Asymptotes of Rational Functions Letr be the rational function Example 3: Graph r(x) = 1. The vertical asymptotes of r are the lines 2. (a) If n<m, then r has (b) If nm, then r has (c) If nm, then r has r(x) = a) Vertical Asymptotes 2x² + 3x - 2 = 0 (2x-1)(x+2) = 0 2x-1= O x = 1/2 x+2=3 x=-2 VA xint y =O ²7-1 +...+q₁x+ao "+an-1-x^² bmx+bm-1x²-1 +…+bx+bo 2x²-4x+5 x²–2x+1 3x²-2x-1=0 (3x+1)(x-1)=0 x = -1/3 >1 b) Horizontal Asymptotes y = 3x²/2x² = 3/2 HA D: x²-2x + 1 = 0 (x-1)² = 0 X 31 D: (-∞0, 1) (1,00) VA: x = 1 HA ⇒ y = 2x² horizontal asymptote y = 0 horizontal asymptote y = a,/b…. no horizontal asymptote and state the domain and range. x² =2 : y = 2 P(2)=8-8 + Example 4: Find the vertical and horizontal asymptotes of x = a, where a is a zero of the denominator range = (2,00) deg of n deg of m y=2 x = 1 r(x) = 3x²2x-1 2x² + 3x-2 k 7 IV. Graphing Rational Functions List the guidelines for sketching graphs of rational functions. 1. Factor. Factor the numerator and denominator. 2. Intercepts. Find the x-intercepts by determining the zeros of the numerator and the y-intercept from the value of the function at x = 0. 3. Vertical Asymptotes. Find the vertical asymptotes by determining the zeros of the denominator, and then see whether y → ∞ or y→→→∞ on each side of each vertical asymptote by using test values. 4. Horizontal Asymptote. Find the horizontal asymptote (if any), using the procedure described earlier. 5. Sketch the Graph. Graph the information provided by the first four steps. Then plot as many additional points as needed to fill in the rest of the graph of the function. Example 5: Graph Step 1: Factor = (2x-1)(x+4) (x+2)(x-1) Step 2: Intercepts r(x) = 2x² +7x-4 x² + x -2 -2,1 D:X-2,1 x-intercepts=0 (2x-1) (x+4)=0 x = 1/2-4 Step 3: Vertical Asymptotes and state the domain and range. y-intercepts=0 y = -4/-2 = 2 8 Behavior near vertical asymptotes. We need to know whether y→∞ or y→∞ on each side of each vertical asymptote. To determine the sign of y for x-values near the vertical asymptotes, we use test values. XB-2 X @ 1 Step 4: Horizontal Asymptotes 2 Step 5: Sketch the graph Domain r(x) = qu ليد- x۲ 2x² +7x-4 x²+x-2 y=2 XB-2 XE 1 Range (0) 9 Is it possible for a graph to cross a vertical asymptote? Is it possible for a graph to cross a horizontal asymptote? V. Common Factors in Numerator and Denominator Example 8: Graph the following X-3 x²-3x (x-3) x(x-3) If s(x) = p(x)/q(x) and if p and q have a factor in common then we may cancel that factor, but only for those values of x for which that factor is not zero (because division by zero is not defined). Since s is not defined at those values of x, its graph has a hole at those points. a) s(x) = D: x 0,3 VA x = 3 hole x = 3 R: y# 0₂3 y. s(x) - x - 3 x² 3.x 1 0 = 1 tions y = 1/x s is not defined for x = 3 s(x) = 1/x for x #3 (a, 1/3) X Do b) yes t(x) = x³ - 2x² X-2 (2,4) hole : 2 y = (2)² =4 t(x) = x³ 0 x2(x-2) (x-2) t is not defined for x = 2 t(x) = x² forx #2 D: (-∞0,2) (2,00) :0,00) (234) X 10 11 VI. Slant Asymptotes and End Behavior If r(x)=P(x)/Q(x) is a rational function in which the degree of the numerator one more than the degree of the denominator we can use the Division Algorithm to express the function in R(x) Q(x), where the degree of R is less than the degree of Q and a *0. For the form r(x) = ax +b + large values of situation we say that y = ax + b is a slant asymptote or an oblique asymptote Factor: Example 9: Graph the rational function f(x) = (x - 5)(x+1) x-3 x-intercepts=0 (x - 5)(x+1)=0 x = 5, -1 the graph of y=(x) approaches the graph of the line y = ax + b Vertical asymptote x=3 Behavior near vertical asymptotes x-3- 849 X-3+ r(x) = x² - 4x-5 X-3 y-intercepts x = 0 y = -5/-3 = 5/3 Horizontal asymptote ×²× =X no HA In this 12 Slant asymptote Graph 314 5 -3 f(x) = x-1 + 8/x-3 Blant y = x-1 y = x 1 x43- y→∞ 5 r(x) = 1-1-3 5/3 2 x-3+ g115 x² - 4x-5 X-3 5 r(x) = Slant asymptote x² - 4x - 5 x 3 X 13