One technique for determining the end behavior of a rational function is to divide each term. To determine the end-behavior of the given rational function, use the table capability of your calculator to determine the limit of the function as x approaches positive and/or negative infinity (as we did in the sequences shown in Figure \(\PageIndex{7}\) and Figure \(\PageIndex{8}\)). Then the rational function n(x) d(x) tends to zero as x grows large in absolute value. Analyzing a rational function (ex. I really do not understand how you figure it out. Linear, quadratic, cubic, and higher order polynomial functions are types of rational functions. is a polynomial function with integral coefficients (a n ≠0 and a 0 ≠0) and (in lowest terms) is a rational zero of ( ), then p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n . Step 3:Look at the end behavior (look at the leading terms of the numerator and denominator)-if there is a higher degree in the denominator--the function is getting closer to zero Step 4: Determine the Horizontal asymptote (from end behavior or dividing the numerator by the denominator after graphing the function) Recall that a polynomial’s end behavior will mirror that of the leading term. x. End behavior of rational functions. What is the end behavior of rational functions? Simply writing a or -1 does not describe a line. What is the function doing on its extremes? Look at graphs: Approaches horizontal asymptote! End behavior:. For example, the function. Use the end behavior and the behavior at the intercepts to sketch the graph. For each function, write “ x-intercepts, y-intercepts, horizontal asymptotes, vertical asymptotes,” and “points of discontinuity” on separate lines below the function. 6 Rational Functions A rational function f(x) is a function which is the ratio of two polynomials, that is, f(x) = n(x) d(x) (18) where n(x) and d(x) are polynomials. We’ll do the middle part next time. With rational functions, end behavior models are determined by inﬁnite limits I need some help with figuring out the end behavior of a Rational Function. vertical asymptotes. The end behavior of a rational function describes how the function f(x) may behave when the input x is a very large positive or negative value. You have already seen some specific types of rational functions. Write a rational function that describes mixing. 3 Properties of Rational Functions 187 1 Find the Domain of a Rational Function Finding the Domain of a Rational Function (a) The domain of is the set of all real numbers x except that is, (b) The domain of is the set of all real numbers x except and 2, that is, (c) The domain of is the set of all real numbers. See [link], [link], [link], An oblique (or slant) asymptote is another type of end behavior for rational functions. By looking at the graph and substituting a few successively larger values of 𝑥 into the function, what is the end behavior of the graph as 𝑥 increases along the positive 𝑥-axis? (a) The value of 𝑦 approaches infinity Section Long-Run Behavior of Rational Functions. Use long division to divide the. Graphs of Rational Functions Name_____ Date_____ Period____-1-For each function, identify the points of discontinuity, holes, intercepts, horizontal asymptote, domain, limit behavior at all vertical asymptotes, and end behavior asymptote. Learn how to visualize and find the oblique asymptotes of a rational function. and then the asymptote would be End Behavior: The end behavior of a graph of a function is how the graph behaves as x x approaches infinity or negative infinity. contributed. So I was wondering if anybody could help me out. 5 # 24-26 (Use RRT), 27-29. e. Author: Jacob Schares. No When determining end behavior of a function, the only term that matters in the numerator and denominator is the variable term with the largest exponent. 2 End Behavior of Rational Functions. See full list on study. Graph a rational function given horizontal and vertical shifts. •Rational functions behave differently when the numerator isn’t a constant. 0 times. Mathematics. (As with rational numbers, the word rational refers to a ratio. , the end behavior asymptote is the quotient polynomial function where. 10. x 4 are eventually positive to the right(end behavior), so their ratio must be positive to the right. The roots, zeros, solutions, x-intercepts (whatever you want to call them) of the rational function A rational function is a function whose formula can be written as the ratio of two polynomial functions. Sign In Describe the end behavior of the following rational functions. Write the letter of the correct end behavior in each answer box. This lesson begins with a set of exercises that provides an opportunity to Determine the end behavior by examining the leading term. de 2011 LIMITS OF RATIONAL FUNCTIONS AS x →±. Horizontal: If the degree of ( ) The end behavior is same as the end behavior of the polynomial. A vertical asymptote is a value of Rational functions have two categories of asymptote: 1. We have previously seen that a polynomial function is defined for all values of x, x , and its graph is a smooth curve without any breaks or holes. Figure 6 The rational function f(x) = can be transformed by using methods similar to those used to transform other types of functions. I need some help with figuring out the end behavior of a Rational Function. Math Lab: End Behavior and Asymptotes in Rational Functions Cut out the tiles and sort them into the categories below based on their end behavior. Over the past few years of teaching precalculus regularly, I've experimented with a variety of approaches to rational functions in an attempt to find one that This is "SM III 4. END BEHAVIOR OF RATIONAL FUNCTIONS Assumed prior knowledge: a) TI-83 techniques - function graphing and window management - table generation b) Algebra concepts or notation - Division of polynomials to produce a polynomial quotient - Understanding of “ as X approaches a value, the corresponding Y approaches a value. at y = (leading coeff. com Match each rational function with a description of its end behavior as x x x gets larger and larger. The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. Note the vertical asymptote and the intercepts, and how they relate to the function. A rational function is two po lynomials divided by each other. For instance, \(r(x) = \frac{7x^3 - 5x + 16}{-4x^4 + 2x^3 - 11x + 3}\) is a rational function. ) Rational Functions. Examples. de 2018 How do I graph the rational function y=x+2x2+1 on a graphing calculator? How do you find the inverse function of: y= Lines that the graph of a rational function approach. The graph of Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. An oblique asymptote refers to "end behavior like a line with nonzero slope," Rational Functions - More on Limits · Remember end-tail behavior? · This was called "finding limits. More examples can be seen here. Section Short-Run Behavior of Rational Functions Subsection Vertical Asymptotes and Holes. For sinx x the limit as it approaches 0 is 1 (proof too hard), and as it approaches infinity: © 2003-2021 Chegg Inc. However, there is a nice fact about rational functions that we can use here. Then sketch the graph. The function 𝑓𝑥=1 𝑥 has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. H. End behavior of a polynomial: always goes to . Real-life Applications Rational functions are useful as examples of graphs that have many interesting features, such as asymptotes and non-obvious intervals of increasing and decreasing. A rational function is a fraction of polynomials. Rational Function: a type of function containing two polynomial functions Step 1: Look at the zeros of the Denominator-zeros are the vertical asymptote(s) of the functionStep 2: Look at the zeros of the numeratorStep 3:Look at the end behavior (look at the leading terms of the numerator and denominator)-if there is a higher degree in the End behavior of a rational function calculator. f(x)=\frac{x^{5}-1}{x+2} Monique R. Determine the end behavior of the following rational functions. Log InorSign Up. )/(leading coeff. Identify horizontal and vertical asymptotes of rational functions from graphs. SECTION 3. Check with a classmate before gluing them. First, the right end. A rational function is the quotient of two polynomials. At first glance, these questions about zeros and vertical asymptotes of rational functions may appear to be The end behavior: is the behavior of the graph as x is approaching + ∞ or -∞ The graph of even degree function may or may not intersect the x-axes depending on its location in coordinate plan. The graph extends down as you approach 2 from the left, and it extends up as you approach 2 from the right. de 2012 They have vertical asymptotes (where the denominator polynomial is zero but the numerator polynomial is not zero), and · They have end-behavior It is only the end behavior of the graph of a rational function that is determined by the horizontal or slant asymptote. For example, (1) is a rational function. 3 cases of end behavior. A rational function is a function that can be written as the ratio of two polynomials where the denominator isn't zero. (a) f(x) = 3x 1 2 5x (b) g(x) = 2x x+7 (c) h(x) = x+7 x2 6x+8 for the following rational functions Arron Kau. What is the end behavior of this rational function? If you are interested in the end behavior, you are concerned with very, very large values of x. " · One of the things math geeks get all jazzed about in 31 de ago. The second part of the activity (not shown) asks them complete three statements: If the degree of the numerator is greater than the degree of the denominator, then…. Graphing a Rational Function with H. In this case, both the numerator and denominator are quadratic polynomials. This can sometimes save time in graphing rational functions. When determining end behavior of a function, the only term that matters in the numerator and denominator is the variable term with the largest exponent. Polynomials - End Behavior Describe the end behavior of each function. Time-saving video on how to determine end behavior and Calculate the end behavior of a rational function by rewriting it in the form . 2) Analyzing a rational function (ex. using ALEKS. Use that information to sketch the graph. I. That is, if p(x)andq(x) are polynomials, then p(x) q(x) is a rational function. A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. 1) f (x) = x3 + 10 x2 + 32 x + 34 f (x) → −∞ as x → −∞ f (x) → +∞ as x → +∞ 2) f (x) = −x2 − 8x − 15 f (x) → −∞ as x → −∞ f (x) → −∞ as x → +∞ 3) f (x) = −x4 + x2 + 2 f (x) → −∞ as x → −∞ f (x) → −∞ as x Exercise Set 2. coefficient to determine its end behavior. End Behavior of Rational Functions. Since this is true, unlike vertical asymptotes, Rational functions also have end behavior asymptotes. an algebraic fraction such that both the numerator and the denominator are This is the activity I used a couple of months ago to help students investigate the end behavior of rational functions. Also, what makes a function rational? In mathematics, a rational function is any function which can be defined by a rational fraction, i. Today, we will go over a quick analysis of the end behavior for rational functions. You can find it without drawing the graph by dividing the leading term of the polynomial in the numerator by the leading term of the polynomial in the denominator. In a similar way, any polynomial is a rational function SECTION 3. Evaluate the following limits of rational functions: (a)lim x!1 (10 + x 2) (b)lim x!1 3x4 + 10x5 2x7 + 4 (c)lim x!1 5x3 x3 + 2x2 + 1 Problem 6. What is the end behavior of a rational function? When the degree of the numerator is two or more higher than the degree of the denominator, the. For example, f(x) = 3x2 x 4 x2 2x 8 is a rational function. x x - or. •An end-behavior asymptoteis an asymptote used to describe how the ends of a function behave. 4), but also vertical asymptotes. Fresh eyes on end behavior of rational functions. The distance between the curve and the line approaches zero as we move out further and further out on the line. An intercept of a rational function is a point where the graph of the rational function intersects the. As the name suggests, end behavior asymptotes model the behavior of the function at the left and right ends of the graph. A few rational function End Behavior of Rational Functions. The horizontal asymptotes of a function correspond to the "end behavior" or the limit of the function as x!1or x!1 . These are called limits AT in nity because xis going to in nity. Suppose that f(x) = n(x) d(x) is a rational function where the taught end behavior and domain and range, have students complete the Extension exercise. Vertical asymptotes: These occur at the real 12 de abr. • 3(x5) (x1) • 1 x • 2x 3 1 =2x 3 The last example is both a polynomial and a rational function. We have previously seen that a polynomial function is defined for all values of \(x\text{,}\) and its graph is a smooth curve without any breaks or holes. 3: Rational Functions MATH 1330 Precalculus 229 Recall from Section 1. 2 that an even function is symmetric with respect to the y-axis, and an odd function is symmetric with respect to the origin. A rational function is one of the form © 2003-2021 Chegg Inc. 21 de set. Like logarithmic and exponential functions, rational functions may have asymptotes. End behavior of a graph describes the values of the function as x approaches positive infinity and negative infinity positive infinity goes to the right x o f negative infinity x o f goes to the left Identifying Horizontal Asymptotes of Rational Functions. Classwork Opening Exercise (3 minutes) The work in Algebra II showed students how to analyze the end behavior of polynomials. That is, if r(x) is Generalized Technique for the end behavior of a rational function: To deter-. 1. It can be written in the form where P(x) and Q(x) are polynomial functions, f(x) — and Q(x) O. Problem 5. The y = the quotient is the function which is the end behavior asymptotic curve for any rational How to analyze end behavior of limits and evaluate the limit of a function as it approaches infinity. y = ( x + 2) ( x − 1) ( x − 3) For each function, find the x-intercepts, y-intercept, vertical asymptotes, end behavior (horizontal asymptotes), and holes if applicable. at y = 0. A few rational function problems: Find the intercepts and asymptotes (vertical, horizontal, or slant) of each of Improve your math knowledge with free questions in "Determine end behavior of polynomial and rational functions" and thousands of other math skills. What is the limit of function e as x approaches negative infinity? Sect. If a rational function has x- In this video, we will study the end behavior of rational functions. Rational Roots Theorem. •It is possible to determine these asymptotes without much work. With rational functions, end behavior models are determined by inﬁnite limits Question Video: Evaluating the End Behavior of Rational Functions. The end behavior of a function Unit 8 -Lesson 3 – End behavior of rational functions. Generalize from specific rational functions to state relationships between the Recall that a polynomial's end behavior will mirror that of the leading term. Solving Polynomial Equations by Factoring. A rational function has a slant asymptote if the degree of a numerator polynomial is 1 more than the degree of the denominator polynomial. If. See Example , Example , Example , and Example . Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. Then, determine their x-intercepts, y-intercepts, and horizontal/vertical asymptotes, if there are any. Pictured below is the graph of a polynomial. A. Determine end behavior by dividing and seeing what terms End Behavior of Rational Functions to rewrite a variety of different rational expressions in order to identify the end behavior of the function. y. There are three distinct outcomes when checking for horizontal asymptotes: A rational function is a fraction of polynomials. Rational Functions, A rational function is simply the ratio of two polynomial with theoretical/asymptotic behavior outside the domain of interest. A few rational function problems: Find the intercepts and asymptotes (vertical, horizontal, or slant) of each of I don’t see that the linear term has much to do with asymptotic behavior of a rational function except in the case when the numerator has degree one greater than the denominator. 4) Analyzing a rational function – slant asymptote. they graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior (F-IF. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. It is helpful when you are graphing a polynomial function to know about the end behavior of the function. As the values of x x approach infinity, the function values approach 0. 6. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are avail-able, and showing end behavior. What happens as x increases without bound? Preliminaries: Illustrate the four types of end behaviors for polynomials. A rational function is one of the form 2. 1 Algebra with mixed fractions For rational functions this may seem like a mess to deal with. In a similar way, any polynomial is a rational function To determine the end-behavior of the given rational function, use the table capability of your calculator to determine the limit of the function as x approaches positive and/or negative infinity (as we did in the sequences shown in Figure \(\PageIndex{7}\) and Figure \(\PageIndex{8}\)). asymptotes which determine the end behavior - these could be either Horizontal asymptotes of rational functions will help you describe the end behavior of a graph. C. Step 3:Look at the end behavior (look at the leading terms of the numerator and denominator)-if there is a higher degree in the denominator--the function is getting closer to zero Step 4: Determine the Horizontal asymptote (from end behavior or dividing the numerator by the denominator after graphing the function) If the degree of the numerator is exactly one more than the degree of the denominator, the end behavior of this rational function is like an oblique linear function. An example of a rational function would be . Khan Academy is a 501(c)(3) nonprofit organization. A rational function will be zero at a particular value of \(x\) only if the numerator is zero at that \(x\) and the denominator isn’t zero at that \(x\). The graph of the rational function attempts to imitate an oblique straight line in the domain's positive or negative extremes. an algebraic fraction such that both the numerator and the denominator are Rational Function: a type of function containing two polynomial functions Step 1: Look at the zeros of the Denominator-zeros are the vertical asymptote(s) of the functionStep 2: Look at the zeros of the numeratorStep 3:Look at the end behavior (look at the leading terms of the numerator and denominator)-if there is a higher degree in the END BEHAVIOR OF RATIONAL FUNCTIONS Assumed prior knowledge: a) TI-83 techniques - function graphing and window management - table generation b) Algebra concepts or notation - Division of polynomials to produce a polynomial quotient - Understanding of “ as X approaches a value, the corresponding Y approaches a value. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonomet-ric functions, showing period, midline, and amplitude. Instead of the ends approaching a horizontal line, the ends approach Consider the three rational functions whose graphs are shown below. \quad\quad b) The value of the expression gets closer and closer to 1. Two aspects of rational functions are straightforward to determine for any rational function. de 2015 Use arrow notation to describe the end behavior and local behavior of the function graphed in Figure 6. Recall that a polynomial’s end behavior will mirror that of the leading term. Suppose that f(x) = n(x) d(x) is a rational function where the I don’t see that the linear term has much to do with asymptotic behavior of a rational function except in the case when the numerator has degree one greater than the denominator. a) The value of the expression gets closer and closer to 0. Topic: Functions. To find the end behavior model for a rational function, use the ratio of the end behavior models for each polynomial. As x gets very, very large, the highest degree term becomes the only term of interest. Page 7. 1 x Slide 2 of 25. →−∞, →0 →∞, →0 →−∞, →−∞ →∞, →∞ OR →−∞, →∞ →∞, →−∞ →−∞, →𝑐 This function, y = 1, is called the end behavior model function for the rational function on the graph. If the degree of the numerator is equal to the degree of Use arrow notation to describe local and end behavior of rational functions. Fun maths practice! Improve your skills with free problems in 'Determine end behaviour of polynomial and rational functions' and thousands of other practice lessons. 5. pdf. Examples: 1. as x heads to infinity and as x heads to negative Free Functions End Behavior calculator - find function end behavior step-by-step. Solve applied problems involving rational functions. 12 -Writing Functions Given Zeros-Fundamental Theorem of Algebra-Irrational and Complex Conjugate Roots Theorems Function raph Value, Eve n Range: [o, m) End Behavior: End Behavior: Rational (Inverse Squared), Even Domain: Range: End Behavior: (y = 2 in the graph) 2- End Behavior of Rational Functions Worksheet. Likewise, a rational function's end behavior will mirror that of the ratio of Students explore end behavior of rational functions graphically, algebraically, and by using Graph a rational function to verify its domain and range. There is no horizontal asymptote. Graph of f(x)=1/(x-2)+4. Use arrow notation to describe local and end behavior of rational functions. Students will be able to identify End Behavior of polynomial equation and predict End Behavior based on the degree and leading coefficient of a polynomial equation. While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. 1) f (x) x x y Identifying Horizontal Asymptotes of Rational Functions. This is the activity I used a couple of months ago to help students investigate the end behavior of rational functions. Description. At first glance, these questions about zeros and vertical asymptotes of rational functions may appear to be This function, y = 1, is called the end behavior model function for the rational function on the graph. In addition, from the behavior of simple rational power functions such as \(\frac{1}{x}\text{,}\) we expect that rational functions may not only have horizontal asymptotes (as investigated in Section 5. In calculus terms, the limit as x goes to in nity of n(x) d(x) is zero. Place the attached Rational Functions sheets across the top of the board. The vertical asymptotes are where the function’s y-values shoot o to 1or 1 as xgoes to a value. A slant asymptote, just like a horizontal asymptote, guides the graph of a function only when x is close to but it is a slanted line, i. Analyzing a Rational Function – 3 more examples. ) 1. Multiplicity of Roots 6. Give the possible degree of A rational function's end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. Normally you say/ write this like this. If a function is even or odd, then half of the function can be Plan your lesson in Polynomial and Rational Functions and polynomial expressions with helpful tips from teachers like you. A rational function is a fraction where the numerator and denominator are both polynomials. The End Behavior for rational functions is tricky they can sometimes be described with horizontal asymptotes, when they cannot be described by these they End behavior of a rational function: In a rational function ( f(x) / g(x) = y(x) ) the quotient plus the remainder is what the graph will look like. Polynomials with even degree behave like power If you are concerned by the behavior of the function when x starts to be large, just perform the long division of polynomials. 1) Analyzing a rational function (ex. The roots, zeros, solutions, x-intercepts (whatever you want to call them) of the rational function Exercise Set 2. See (Figure). Determine the end behavior by examining the leading term. •There are two types of end-behavior asymptotes a rational function can have: •(1) horizontal •(2 End behavior: what the function does as x gets really big or small. End behavior of a rational function calculator. Remember that a rational function is a ratio of polynomials, where the polynomial in Calculate the end behavior of a rational function by rewriting it in the form . This Bakpax autogradable standards-aligned Math worksheet covers End Behavior of Rational…. y y -axis. The parent rational function is 𝑓𝑥=1 𝑥. Do the same on the overhead calculator. 11 Rational Roots Theorem and Solving Polynomial Equations with the help of a calculator. DRAFT. End behavior of rational functions Our mission is to provide a free, world-class education to anyone, anywhere. 1 Algebra with mixed fractions Section Long-Run Behavior of Rational Functions. 5 # 2-4, 11-13. All rights reserved. Objectives: Students will be able to: •. f(x) = p(x) / q(x) Domain. A rational function is any function that can be written as the ratio of two polynomials. 3) Analyzing a rational function (ex. In Exercises $57-62$ , find the intercepts, vertical asymptotes, and end behavior asymptote, and graph the function together with its end behavior asymptote. Roots. Over the past few years of teaching precalculus regularly, I’ve experimented with a variety of approaches to rational functions in an attempt to find one that will result in students discovering and eventually truly understanding how the equation of a rational function determines its graph. If the degree of the numerator is equal to the degree of What is the end behavior of a rational function? When the degree of the numerator is two or more higher than the degree of the denominator, the. Generalize from specific rational functions to state relationships between the End behavior is just how the graph behaves far left and far right. This prepares students for subsequent lessons in which they graph rational functions, identifying zeros and asymptotes when suitable End behavior: what the function does as x gets really big or small. 5 Rational Functions End Behavior Part 1" by Mountain Heights Academy Videos on Vimeo, the home for high quality videos Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large . It helps by knowing the limits of the function (eg sinx is between -1 and 1), transforming the simple function to the complex one and, if the side limits are equal, then they squeeze the answer between their common answer. First, the end behavior of a polynomial is determined by its degree and the sign of the lead coefficient. Examples: 1) 4 6 ( ) 2 6 x f x x Ask students to graph the function on their calculators. The arrows indicate the end behavior of the numerator and denominator, so we can quickly know the sign of the rational function left and right. In the middle we can use the fact that an odd exponent signals a sign change, while A rational function is a function thatcan be written as a ratio of two polynomials. If intersects the x-axes in 2 places, the function has two real zeros. e. This lesson offers students opportunities to use tables to analyze the end behavior of rational functions and the behavior of rational functions as they approach restricted input values. the end behavior of the graph of a rational function that is determined by the horizontal or slant asymptote. The numerator is p(x)andthedenominator is q(x). d. end behavior of a rational function is modeled by a polynomial. numerator by the denominator. I looked at this question:How do you determine the end behavior of a rational function? but it made me even more confused on how to figure out the end behavior. Rational Functions What is the limit of function e as x approaches negative infinity? Sect. Example: An end behavior model of 2 32 3 5 7 2 5 4 xx gx x x x is 2 3 33 22 x xx The en d behavior of the “model” is the same as the end behavior of the original A rational function is a function that can be written as the ratio of two polynomials where the denominator isn't zero. The domain of a rational function is all real values except where the denominator, q(x) = 0. For example We'll begin by discussing the “end- behavior” of the rational function defined by y = 1/x. Graph: Y1 = 1 / X 2. neither vertical nor horizontal. There are three distinct outcomes when checking for horizontal asymptotes: the end behavior of the graph of a rational function that is determined by the horizontal or slant asymptote. GeoGebra Applet Press Enter to start activity A rational function is the quotient of two polynomials. Download and share any assignment - for free. Consider the graph of the function 𝑦 = 1/𝑥. 6. 2- End Behavior of Rational Functions Worksheet. The polynomial is y = quotient. The graph of r( 13 de fev. In calculus, you will evaluate limits of rational functions. Smith (SHSU) Elementary Functions 2013 19 / 42 End-behavior of Rational Functions Repeating the previous slide: Lemma 2. (a) y = x+ 3 x+ 5 (b) y The end behavior: is the behavior of the graph as x is approaching + ∞ or -∞ The graph of even degree function may or may not intersect the x-axes depending on its location in coordinate plan. Many rational functions contain vertical asymptotes and end behavior asymptotes. 11th grade. If a function is even or odd, then half of the function can be Fun maths practice! Improve your skills with free problems in 'Determine end behaviour of polynomial and rational functions' and thousands of other practice lessons. 2 End Behavior of Rational Functions Practice. 2. 7d). •To find the rational zeros, divide all the factors of the constant term by all the factors of the lead coefficient.