Sec. 3.7(B) Finding the V.A. , H.A. , X-Intercept, and Y-Intercept of a Rational Function Objective: Students will be able to find the vertical asymptote, horizontal asymptote, x-intercept, and y-intercept of a rational function.

Rational Function:   Domain of a Rational function: **If x = a is an excluded number, then there is either a 1) Or 2) The parent rational function is 𝑓 𝑥 = 1 x . The graph of this function is below: Is a quotient of two polynomials functions of the form 𝑓 𝑥 = 𝑔(𝑥) ℎ(𝑥) , where h(x) ≠0. Is all real numbers that do not make the denominator equal to zero. The numbers that make the denominator equal to zero are excluded numbers from the domain. Vertical asymptote at the line x = a There is a hole at x = a, if the factor x – a cancels in both numerator/denominator.

The graph of 𝑓 𝑥 = 1 x , like that of many rational functions, has branches that approach a line called ____________.   The line x = 0 is a __________________. A vertical asymptote in the graph of a rational function is also called a _______ of the function. Notice that as the x-value approaches zero from the left, the value of f(x) decreases without bound toward ______________________. As the x-value approaches zero from the right, the value of f(x) increases without bound toward ________________________. The line y = 0 is a _____________________. asymptotes vertical asymptote pole negative infinity (−∞) positive infinity (∞) horizontal asymptote On the graph, you notice that as the x values decrease the value of f(x) approaches 0 and also as the x-value increases on the right side the value of f(x) approaches 0.

There are 2 methods you can use to find a Horizontal Asymptote:   1. _____________________________________________________________________ 2. _____________________________________________________________________ Let f(x) = y and solve for x. Find out where the function is undefined, this is the horizontal asymptote. Look at the Exponent in Numerator and compare it to the Exponent in the Denominator Example 1: Determine the asymptotes for the graph of: 𝑓 𝑥 = 𝑥 𝑥−1 .   Vertical Asymptote Since the denominator would be equal to zero at f(1), f(1) is undefined. It is possible to have a vertical asymptote at x = 1. To verify that x = 1 is indeed a V.A., you have to make sure that 𝑓 𝑥 → ∞ 𝑜𝑟 𝑓 𝑥 →−∞ as x →1 from either the left or right. x f(x) 0.9 -9 0.99 -99 0.999 -999

Look at the exponents of the numerator/ denominator. Horizontal Asymptote   Method 1 Method 2 y= 𝑥 𝑥−1 𝑦 𝑥−1 =𝑥 xy – y = x xy – x = y x(y – 1) = y x = 𝑦 𝑦−1 Look at the exponents of the numerator/ denominator. 1. If N > D , no H.A. , but the end behavior of the graph is L.S. (Down) and R.S. (Up). 2. If N = D, then H.A.: y = 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 Undefined for y = 1, so the horizontal asymptote is the line y = 1. 3. If N < D , then H.A.: y = 0.

f(x) = 𝑥−2 𝑥 2 −𝑥−2 Intercepts of a Graph To find the y-intercepts of a graph: Find f(0). 𝑓 0 = 0 0−1 = 0 Y-intercept: (0, 0 ) f(x) = 𝑥−2 𝑥 2 −𝑥−2 EX: f(0) = 0−2 0 2 −0−2 = −2 −2 = 1 So the y-intercept is (0, 1). To find the x-intercepts of a graph: Set the numerator equal to 0 and solve. x = 0 X-intercept is (0, 0) f(x) = 𝑥−2 𝑥 2 −𝑥−2 EX: x – 2 = 0 x = 2 So the x-intercept is (2, 0).

𝑓 𝑥 = 𝑥 2 −4𝑥 𝑥 3 − 𝑥 2 −20𝑥 Hole: f(0) = undefined , x = 0 Example 2: Determine the asymptotes and intercepts of the rational function: 𝑓 𝑥 = 𝑥 2 −4𝑥 𝑥 3 − 𝑥 2 −20𝑥 Factor the numerator and denominator. 𝑓 𝑥 = 𝑥(𝑥−4) 𝑥( 𝑥 2 −𝑥−20) = 𝑥(𝑥−4) 𝑥(𝑥−5)(𝑥+4) H.A. : Numerator Exp. < Denominator Exp. y = 0 Y-Int.: Find F(0) No y-intercept. 𝑓 0 = 0 2 −4(0) 0 3 − 0 2 −20(0) = 0 0 = undefined Hole: f(0) = undefined , x = 0 V.A.: f(5), f(-4) = undefined x=5 , x = -4 x f(x) -4.1 -8.901 -4.01 -88.901 -4.001 -888.901 x f(x) 4.9 -1.011 4.99 -11.012 4.999 -111.012 X-Int.: Set numerator = to 0 & solve 𝑥 2 −4𝑥=0 𝑥 𝑥−4 =0 x = 0 x = 4 (0, 0) is a hole & (4, 0) is x-int.

Sec. 3.7(C) Finding a Slant Asymptote & Sketching a Graph of a Rational Function Objective: Students will be able to find the slant asymptote of a rational function and also graph the rational function.

A third type of asymptote is a _________________ A third type of asymptote is a _________________. Slant asymptotes occur when the degree of the numerator of a rational function is exactly 1 greater than the degree of the denominator.   EX: 𝑥 3 +1 𝑥 2 , degree of numerator is 3 degree of denominator is 2 When the degrees are the same or the denominator is greater, than the function has a ____________________. slant asymptote horizontal asymptote Example 4: Determine the slant asymptote for 𝑓 𝑥 = 4 𝑥 2 +6𝑥−37 𝑥+4 . To find the slant asymptote we need to do long division with the N/D. 4𝑥−10+ 3 𝑥+4       4𝑥−10 𝑥+4 4 𝑥 2 +6𝑥−37               4 𝑥 2 + 16𝑥                      −10𝑥−37                      −10𝑥 −40                                         3  So as x gets greater the fraction 3 𝑥+4 approaches 0. This means that the line y = 4x – 10 is the slant asymptote for the graph of f(x).

There are times when the numerator and denominator of a rational function share a common factor.   EX: 𝑓 𝑥 = 𝑥 3 +3 𝑥 2 −4𝑥−12 𝑥+3 = 𝑥 2 −4 𝑥+3 𝑥+3 = (𝑥+2)(𝑥−2)(𝑥+3) 𝑥+3 = 𝑥 2 - 4 (A hole at x = -3) Whenever the numerator and denominator of a rational function contain a common linear factor, you would think you have a vertical asymptote at the number that makes the denominator zero, but you don’t, you may have ___________________ (hole) in the graph of the function. If, after dividing the common linear factors, the same factor remains in the denominator, a _________________ exists.   𝑓 𝑥 = 𝑥+3 𝑥 2 +6𝑥+9 point discontinuity vertical asymptote Since the factor (x+3) was in N/D, but the x+3 still remains In the denominator, x = -3 is a V.A. = 𝑥+3 𝑥+3 2 = 1 𝑥+3

Example 2: Use the parent graph, 𝑓 𝑥 = 1 𝑥 , to graph each equation Example 2: Use the parent graph, 𝑓 𝑥 = 1 𝑥 , to graph each equation. Describe the transformations that have taken place. Identify the new locations of the asymptotes. a) k(x) = 1 𝑥−3 +2 f(x) = − 3 2𝑥−4 −1

𝑐) 𝑓 𝑥 = 3𝑥−1 𝑥−2

Example 3: Sketch the graphs of the following rational functions. b. 𝑓 𝑥 = 3 𝑥 2 −3𝑥 𝑥 2 +𝑥−12

Example 3: Graph 𝑓 𝑥 = (𝑥+2)(𝑥−3) 𝑥 𝑥−4 2 (𝑥+2) .

Example 4: Graph: 𝑓 𝑥 = 𝑥 3 −1 𝑥 2 −9

Many real-world problems can be modeled by rational functions.   Example 3: In an electronics laboratory, an engineer determined that the current l in an electric circuit was 𝑙 𝑥 = −2 𝑥 2 +10𝑥+1 5−𝑥 , where x represents the time that it takes for the current to pass through a circuit. a) Write the function as a transformation of the parent graph of 𝑓 𝑥 = 1 𝑥 . b) Graph the function and identify the asymptotes. Interpret their meaning in terms of the problem. Do long division to put the function in the form to show transformations. 𝑓 𝑥 = 2𝑥+ 1 5−𝑥 V.A.: 5 – x = 0 -x = -5 x = 5 H.A.: Exception – Since x can be anything we have no H.A. y-int.: f(0) = 2 0 + 1 5−0 = 1 5