Rational Functions Notes SOLUTIONS
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MATH ALGERBRA
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Mathematics
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Dec 22, 2024
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Uploaded by PrivateViperMaster1185
IB Analysis and Approaches HL2Chapter 6DRational Functions (SOLUTIONS)Degree of a polynomialRecall that the degree of a polynomial is the highest power to which the xterm is raised.Rational FunctionsA rational function is a ratio of polynomials where the denominator polynomial has a degree greater than or equal to 1. In this lesson, we will only study rational functions with polynomials of degree 1 or 2.Rational functions are usually characterized by the presence of at least one vertical asymptote, at which the value of yis undefined because plugging in the xvalue(s) of the asymptote(s) results in a fraction of denominator 0, which is always undefined.Example 1Consider the rational function y=2x−6x2−3x−4.a)Find the vertical asymptotes.b)Find the horizontal asymptote, if it exists. Try taking the limit of the functionc)Find the axis intercept(s) of the function.d)Draw a sign diagram of the function on the x-axis of the given graph paper.e)Graph the function showing these features. a)x2−3x−4=(x+1) (x−4)∴x=−1, x=4∴vertical asymptotes at x=−1, x=4b)The denominator will be exponentially larger than the numerator, so the ratio gets smaller and smaller until it approaches 0c)0=2x−6x2−3x−4; 0=2x−6;2x=6;x=3; →(3,0)y=2(0)−6(0)2−3(0)−4=−6−4=32; →(0,1.5)
X-InterceptsAn x-intercept occurs when y=0. For a rational function, the only way ycan be 0 is if the numerator also equals 0. Hence, the x-intercept(s) are the zeroes of the numerator.Y-InterceptA y-intercept occurs when x=0. For a rational function, there is always one y-intercept unlessx=0is one of the vertical asymptotes of the function.Vertical AsymptotesA vertical asymptote occurs when yis undefined. For a rational function, yis undefined when the denominator equals 0. Hence, the vertical asymptote(s) are the zeroes of the denominator. If the denominator does not have real zeroes, there are no horizontal asymptotes.Horizontal AsymptotesA horizontal asymptote occurs at limx→∞y. As x→∞, the highest degree term “overpowers” the other terms because it increases at an exponentially higher rate. a)When the numerator’s degree is greater than the denominator degree, the horizontal asymptote does not exist but there is an oblique asymptote.b)When the numerator's degree is lessthan the denominator’s degree, the horizontal asymptote is always at y = 0.c)When the numerator’s degree is equalto the denominator’s degree, the horizontal asymptote is the ratio of the leading coefficients.Oblique AsymptotesTake a rational function in the form y=a x2+bx+cdx+e, such that the numerator has a degree of 2 and the denominator has a degree of 1. Using what you learned from previous lessons, you can divide the polynomial to get the functiony=px+q+rdx+e.
The vertical asymptote is x=−ed. As x→∞, rdx+eapproaches 0, so yapproaches px+q, the oblique asymptote of the function.Example 2Consider the rational function y=x2−3x−42x−6.a)Find the vertical asymptotes.b)Identify the non-vertical asymptote of this function, if it exists. c)Find the line of this asymptote.d)Find the axis intercept(s) of the function.e)Draw a sign diagram of the function on the x-axis of the given graph paper.f)Graph the function showing these features. a)2x−6=0; 2x=6; x=3b)Numerator degree is 2, denominator degree is 1. No horizontal asymptote: instead, there is an oblique asymptotec)From 12x−42x−6, 12xis the line of the oblique asymptote.d)X-intercept: x2−3x−4=0;(x−4) (x+1)=0; x=4,−1Y-intercept: 02−3(0)−42(0)−6=−4−6=(0,−23)