Rational Functions and inequalities

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Example 1*: Find the Domain of a Rational Function

Find the domain of the following rational functions

Example 2: Graph the following functions


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Section 4.2, 4.3 & 4.4 Section 4.2: Properties of Rational Functions https://mediaplayer.pearsoncmg.com/assets/1NIzrJsdy0qx5jwM_vgh0NyjCAMp7Iuu Author in Action: Find the Domain of a Rational Function (5:56) Post 1 Definition: A rational function is a function of the form R ( x ) = where p and q are polynomial functions and q is not the ___________________. The domain of a rational function is _______________________ except those for which the denominator q is ____. WARNING: The domain of a rational function must be found before writing the function in its lowest terms. Example 1*: Find the Domain of a Rational Function Find the domain of the following rational functions x2 − 4 x+6 (a)* R ( x ) = (b)* R ( x ) = 2 x+4 x + 8 x + 12 (c)* R ( x ) = x −5 x2 + 2 (d) R ( x ) = x2 − 9 3 (e) R ( x ) = x2 − 4 x+2 End of Post 1 https://mediaplayer.pearsoncmg.com/assets/EKTJNnlu3ciT871DoeoWkYscimlBFGJD Author in Action: Graph Transformations of a Rational Function (18:10) Example 2: Graph the following functions 1 y= x y= 1 x2 Copyright © 2016 Pearson Education, Inc. Rational Functions In Section 2.4 we created our library of functions and in Section 2.5 we transformed those basic 1 functions. We can do the same types of transformations with the basic rational functions y = and x 1 y= 2 . x Post 2 Example 3*: Graph Transformations of a Rational Function 1 + 2 using transformations. Graph the rational function R ( x ) = 2 ( x − 3) End of Post 2 Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 Long-run behavior: As we learned in Section 51, the long-run behavior of a function concerns what happens as the inputs get extreme ( 𝑥 → ±∞). As we saw before with polynomials, the leading terms of 𝑝(𝑥) and 𝑞(𝑥) determine the long run behavior of r ( x ) . Exploration 1: Long-Run Behavior and Vertical Asymptotes 3x + 1 1. Use a graphing utility to graph f ( x ) = . x +1 (a) What is happening to the function’s values as the x values move towards positive infinity? How can you use mathematical notation to write this? (b) What is happening to the function’s values as the x values move towards negative infinity? How can you use mathematical notation to write this? (c) Look at the equation for 𝑓(𝑥). Can you explain why the function’s values get closer and closer to the values found above as the x values approach positive and negative infinity? Copyright © 2016 Pearson Education, Inc. Rational Functions Know these definitions. Horizontal and Vertical Asymptotes Let R denote a function. If, as _________ or as ________, the values of R ( x ) approach some fixed number L, then the line _______ is a horizontal asymptote of the graph of R. [Refer to (a) and (b) below]. If, as x approaches some number c, the values ________________________, then the line __________ is a vertical asymptote of the graph of R. The graph of R never intersects a vertical asymptote. [Refer to (c) and (d)]. Can the graph of a function intersect a horizontal asymptote? (d) What is the equation for the horizontal asymptote of f ( x ) ? (e) What is/are the equation(s) of the vertical asymptote(s) of f ( x ) ? 4 x2 . x3 − 1 (a) What is happening to the function’s values as the x values move towards positive infinity? How can you use mathematical notation to write this? 2. Use a graphing utility to graph h ( x ) = Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 (b) What is happening to the function’s values as the x values move towards negative infinity? How can you use mathematical notation to write this? (c) What is the equation for the horizontal asymptote of h ( x ) ? (d) Look at the equation for h ( x ) . Can you explain why the function’s values get closer and closer to the values found above as the x values approach positive and negative infinity? (e) What is/are the equation(s) of the vertical asymptote(s) of h ( x ) ? 3. Use a graphing utility to graph the functions below and fill in the table. Function Leading Leading Term Equation of Equation(s) Term in the in the the Horizontal of the Numerator Denominator Asymptote Vertical Asymptote(s) 3x + 1 f ( x) = x +1 x+2 q ( x) = 2 2x − 2 s ( x) = 2 x 2 + 3x − 2 x 2 + 3x + 2 g ( x) = −3x 2 − 2 x + 1 x3 − 1 4. Some patterns/relationships exist in the table above that will help us determine the horizontal and vertical asymptotes of a rational function. What patterns do you notice? Copyright © 2016 Pearson Education, Inc. Rational Functions https://mediaplayer.pearsoncmg.com/assets/DbAIQTeP0xB9F6j8PfcR36YRdHnpX1vt Author in Action: Find Vertical Asymptotes of a Rational Function (8:41) Post 3 p ( x) , in lowest terms, will have q ( x) a vertical asymptote at ______ if r is a ___________________ of the denominator q. Locating Vertical Asymptotes Theorem: A rational function R( x) = What do we mean by lowest terms in the theorem above? Multiplicity and Vertical Asymptotes: If the multiplicity of the zero that gives rise to the vertical asymptote is odd, the graph approaches + on one side of the asymptote and − on the other side. If the multiplicity is even, the graph approaches either + or − on both sides of the vertical asymptote. We will use this for graphing rational functions in the next section. Example 4*: Find Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of each rational function. 5x2 x −3 R ( x) = H ( x) = 3+ x ( x + 2)( x − 2) F ( x) = x −1 2 x + 5x + 4 G ( x) = x 2 + 3x + 2 x2 − 4 End of Post 3 Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 https://mediaplayer.pearsoncmg.com/assets/k0guNLSAV9aqwcIOsZ6kPBGQSnZRJlUF Author in Action: Find Horizontal or Oblique Asymptotes of a Rational Function (20:00) Know this chart. Finding a Horizontal Asymptote or Oblique Asymptote of a Rational Function Consider the p( x) an x n + an −1 x n −1 + ... + a1 x + a0 = rational function R( x) = in which the degree of the numerator is n q( x) bm x m + bm−1 x m−1 + ... + b1 x + b0 and the degree of the denominator is m. 1. If n  m , the line ____________ is a horizontal asymptote. 2. If n = m , the line ____________ is a horizontal asymptote. (That is, the horizontal asymptote is y equals “the ratio of the leading coefficients.”) 3. If n = m + 1, the line ____________ is an oblique asymptote (sometimes called a slant asymptote since it is a slanted line), which is the quotient found using long division. 4. If n  m + 2 , there are ___________________________. The end behavior of the graph will a resemble the power function y = n x n − m . bm Note: a rational function will never have both a horizontal asymptote and an oblique asymptote, but may have neither. Example 5: Find Horizontal Asymptotes of a Rational Function Create a rational function that satisfies the following, and then compare your examples with your classmates. (a) Has a horizontal asymptote at y = 5 . (b) Has a horizontal asymptote at y = 0 . (c) Does not have a horizontal asymptote. Copyright © 2016 Pearson Education, Inc. Rational Functions Post 4 Example 6*: Find Horizontal or Oblique Asymptotes of a Rational Function Find the horizontal asymptote, if any, of each of the following graphs: x+3 2x2 + 7 x −1 R ( x) = 2 R( x) = x + 2x + 5 x2 + 2 R( x) = 5 x 6 − 4 x3 + 3 2 x 6 − 5 x5 + 8 x 4 − 7 x3 + 2 R( x) = x2 + 4 x + 1 x−2 Example 7*: Find Horizontal or Oblique Asymptotes of a Rational Function x2 + 4 x + 1 Find the oblique asymptote of R( x) = . x−2 End of Post 4 Example 8: Find Horizontal or Oblique Asymptotes of a Rational Function 2 x 2 − 3x + 2 Find the horizontal or oblique asymptote, if any, of R( x) = . x −1 Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 https://mediaplayer.pearsoncmg.com/assets/3MrVBoKzmDaPIAyoLB7If0jI1HxuJaNI Analyze the Graph of a Rational Function (20:00) Know these steps. Section 4.3: The Graph of a Rational Function Steps for Analyzing the Graph of a Rational Function Step 1: Factor the __________________________ of R. Find the _____________. Step 2: Write R in ____________ (cancel out anything, if you can). If you can cancel out any terms these are the holes of the function. Step 3: Locate the ____________ of the graph. Determine the behavior of the graph of R at each xintercept, based on multiplicity. Step 4: Determine the ___________________ based on the factors of the denominator of R in lowest terms. Graph each vertical asymptote using a dashed line. Determine the behavior of the graph of R on either side of the vertical asymptote based on the following: • • If the multiplicity of a factor is odd, the graph will approach __________ infinities at the vertical asymptote. If the multiplicity of a factor is even, the graph will approach the ________ infinity at the vertical asymptote . Example of opposite infinities: Example of same infinities: Step 5: Determine the __________________ or _________________, if one exists. Determine points, if any, at which the graph of R ____________ this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph of R intersects the asymptote. Step 6: Use the ______________ of the numerator and denominator of R to divide the x – axis into intervals. Determine where the graph of R is __________ or ___________ the x – axis by choosing a number in each interval and evaluation R there. Plot the points found. Step 7: Use the results in Steps 1 through 6 to graph R. Copyright © 2016 Pearson Education, Inc. Rational Functions Post 5 Example 1*: Analyze the Graph of a Rational Function x2 − 4 Analyze the graph of the rational function R( x) = 2 . x + 3x − 4 End of Post 5 Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 https://mediaplayer.pearsoncmg.com/assets/9hxbcRldmBko5upeCuj5cfcxS5yKlJQW Analyze the Graph of a Rational Function with a Hole (18:19) Look through these examples. Example 2*: Analyze the Graph of a Rational Function with a Hole x2 − 9 Analyze the graph of the rational function R( x) = 2 . x + 9 x + 18 Copyright © 2016 Pearson Education, Inc. Rational Functions Example 3: Analyze the Graph of a Rational Function x 2 + 3x + 2 Analyze the graph of the rational function R( x) = . x Example 4: Analyze the Graph of a Rational Function x 2 − x + 12 Analyze the graph of the rational function R ( x ) = . x2 −1 Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 Example 5: Analyze the Graph of a Rational Function x 2 − x + 12 Analyze the graph of the rational function R ( x ) = . x2 −1 Example 6: Analyze the Graph of a Rational Function 2 x − 1)( x + 2 ) ( Analyze the graph of the rational function R ( x ) = . x( x − 4)2 Copyright © 2016 Pearson Education, Inc. Rational Functions Example 7: Constructing a Rational Function from its Graph Find a formula for the rational function graphed below. Example 8: Solve Applied Problems Involving Rational Functions The concentration C of a certain drug in a patient’s blood stream t minutes after injection is given by 2(1 + 20t ) C (t ) = 2 . t + 30 (a) Find the horizontal asymptote of C (t ) . Interpret this horizontal asymptote in the context of the problem. (b) Using your graphing utility, graph C = C (t ) . (c) Determine the time at which the concentration is highest. Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 Section 4.4: Polynomial and Rational Inequalities There are two ways we can solve polynomial and rational inequalities: graphically and algebraically. Let’s use our knowledge from solving quadratic inequalities in Chapter 3 as well as the information we’ve learned in the previous sections to solve these types of inequalities. First, let’s make sure we understand what certain notation means. Exploration 1: Function Inequalities What does f ( x)  0 mean? What would these solutions look like on the graph of f ( x ) ? What does f ( x )  0 mean? What would these solutions look like on the graph of f ( x ) and how do they differ from the solutions to f ( x)  0 ? What does f ( x )  0 mean? What would these solutions look like on the graph of f ( x ) ? What does f ( x )  0 mean? What would these solutions look like on the graph of f ( x ) and how do they differ from the solutions to f ( x )  0 ? https://mediaplayer.pearsoncmg.com/assets/PzGFK_4KIcET0Lcu0zBKqNXcLH8jO08d Solve a Polynomial Inequality (19:36) Example 1*: Solve a Polynomial Inequality Using a Graph Solve ( x + 3)2 ( x − 1)( x − 4)  0 graphically. Copyright © 2016 Pearson Education, Inc. Rational Functions Copyright © 2016 Pearson Education, Inc. Section 4.2, 4.3 & 4.4 https://mediaplayer.pearsoncmg.com/assets/oLfpBBhuc4APHwZ_jXTs2pZCtXWypKjQ Author in Action: Solve a Rational Inequality (20:30) Example 2*: Solve a Rational Inequality Using a Graph x−4  2 graphically. Solve x+2 Know these steps. Steps for Solving Polynomial and Rational Inequalities Algebraically Step 1: Write the inequality so that a polynomial or rational expression is on the ________ side and the zero is on the right side in one of the following forms: _________ _________ _________ _________ For rational expressions, be sure the left side is written as a _____________________ then find the domain of f. Step 2: Determine the real numbers at which the expression f ___________, and if the expression is rational, the real numbers at which the expression f is ____________. Step 3: Use the numbers found in Step 2 to separate the real number line into intervals. Step 4: Select a number in each interval and evaluate f at the number. (a) If the value of f is positive, then __________________ for all numbers x in that interval. (b) If the value of f is negative, then __________________ for all numbers x in that interval. Copyright © 2016 Pearson Education, Inc. Rational Functions Now that we have steps for solving polynomial and rational inequalities algebraically, let’s go back and solve the inequalities in Examples 1 and 2 using algebra. Example 3*: Solve a Polynomial Inequality Algebraically Solve ( x + 3)2 ( x − 1)( x − 4)  0 algebraically. Example 4*: Solve a Rational Inequality Algebraically x−4  2 algebraically. Solve x+2 Copyright © 2016 Pearson Education, Inc.
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Section 4.2, 4.3 & 4.4
Section 4.2: Properties of Rational Functions
https://mediaplayer.pearsoncmg.com/assets/1NIzrJsdy0qx5jwM_vgh0NyjCAMp7Iuu
Author in Action: Find the Domain of a Rational Function (5:56)
Post 1
Definition: A rational function is a function of the form 𝑅(𝑥 ) =

p
𝑞

where p and q are polynomial

functions and q is not the zero function. The domain of a rational function is
_______________________ except those for which the denominator q is _0.___.
WARNING: The domain of a rational function must be found before writing the function in its lowest
terms.
Example 1*: Find the Domain of a Rational Function
Find the domain of the following rational functions
𝑥2 − 4
(
)
(𝑎)* 𝑅 𝑥 =
𝑥+4
(-∞, -4) U (-4, ∞)

(𝑐)* 𝑅 (𝑥 ) =

𝑥−5
𝑥2 + 2

(𝑏)* 𝑅(𝑥 ) =

𝑥+6
𝑥 2 + 8𝑥 + 12

(-∞, -6) U (-6, -2)U (-2, ∞)

(𝑑) 𝑅(𝑥 ) =

(-∞, ∞)

𝑥2 − 9
3

(𝑒) 𝑅(𝑥 ) =

(-∞, ∞)

(-∞, -2) U (-2, ∞)

End of Post 1

https://mediaplayer.pearsoncmg.com/assets/EKTJNnlu3ciT871DoeoWkYscimlBFGJD
Author in Action: Graph Transformations of a Rational Function (18:10)
Example 2: Graph the following functions
1
y=
x

𝑥2 − 4
𝑥+2

y=

1
x2

Copyright © 2016 Pearson Education, Inc.

Rational Functions

In Section 2.4 we created our library of functions and in Section 2.5 we transformed those basic
1
functions. We can do the same types of transformations with the basic rational functions y = and
x
1
y= 2 .
x

Post 2
Example 3*: Graph Transformations of a Rational Function
1
+ 2 using transformations.
Graph the rational function R ( x ) =
2
( x − 3)

End of Post 2
Copyright © 2016 Pearson Education, Inc.

Section 4.2, 4.3 & 4.4
Long-run behavior: As we learned in Section 51, the long-run behavior of a function concerns what
happens as the inputs get extreme ( 𝑥 → ±∞). As we saw before with polynomials, the leading terms
of 𝑝(𝑥) and 𝑞(𝑥) determine the long run behavior of r ( x ) .
Exploration 1: Long-Run Behavior and Vertical Asymptotes
3x + 1
1. Use a graphing utility to graph f ( x ) =
.
x +1
(a) What is happening to the function’s values as the x values move towards positive infinity?
How can you use mathematical notation to write this?
The function’s values approach to the number 3.
𝑎𝑠 𝑥 → ∞, 𝑓 (𝑥) → 3
(b) What is happening to the function’s values as the x values move towards negative infinity?
How can you use mathematical notation to write this?
The function’s values approach to 3.
𝑎𝑠 𝑥 → −∞, 𝑓 (𝑥 ) →3
(c) Look at the equation for 𝑓(𝑥). Can you explain why the function’s values get closer and
closer to the values found above as the x values approach positive and negative infinity?
Because y = 3 is the horizontal asymptote of the function.

Copyright © 2016 Pearson Education, Inc.

Rational Functions
Know these definitions.

Horizontal and Vertical Asymptotes
Let R denote a function.
If, as x approaches infinity or as x approaches negative infinity, the values of R( x) approach some
fixed number L, then the line y = L is a horizontal asymptote of the graph of R. [Refer to (a) and (b)
below].
If, as x approaches some number c, the values increase or decrease indefinitely , then the line y = c is a
vertical asymptote of the graph of R. The graph of R never intersects a vertical asymptote. [Refer to
(c) and (d)].

Can the graph of a function intersect a horizontal asymptote?
Yes.
(d) What is the equation for the horizontal asymptote of f ( x ...


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