Algebra 2: Chapter 8 Part I Practice Quiz Unofficial Worked-Out Solutions

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Juliet Morrison
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1 Algebra 2: Chapter 8 Part I Practice Quiz Unofficial Worked-Out Solutions In working with rational functions, I tend to split them up into two types: Simple rational functions are of the form y = a x h + k or an equivalent form that does not contain a polynomial of degree higher than 1 (i.e., no x 2, x 3, etc. just x s and constants). Problems 1 and 2 below are of this type. General rational functions are of the form y = P(x) Q(x) where either P(x) or Q(x) is a polynomial of degree 2 or higher (i.e., contains an x 2, x 3, etc.). Problems 3 to 6 are of this type. In general, I like to find the asymptotes for a function first, and then find points that help me graph the curve. The domain and any holes typically fall out during this process. The range and the end behavior become identifiable once the function is graphed. Here we go! SIMPLE RATIONAL FUNCTIONS First, a couple of thoughts about simple rational functions. If you can put a rational function in the form y = a + k, here s what you get: x h Vertical Asymptote: Occurs at x = h. Generally I find the vertical asymptote first. It s easy to find because it occurs at x = h. The reason for this is that x = h gives a denominator of h h = 0, and you cannot divide by zero. Hence, as your x-values approach h, the a denominator of becomes very small, and the graph shoots off either up or down. x h Horizontal Asymptote: Occurs at y = k. The function cannot have a value of y = k because a that would require the lead term, to be zero, which can never happen since a 0. x h Hence, the function can approach y = k, but will never reach it. Domain: All Real x h. No value of x exists at a vertical asymptote. Range: All Real y k. No value of y exists at a horizontal asymptote in simple rational functions. Holes: None. End Behavior: Both ends of the function tend toward the horizontal asymptote, so: As x, y k and As x, y k With this background, let s do problems 1 and 2.

Page 2 1) y = x 1 2 First, note that h = 1 and k = 2 Recall that the simple rational a form is: y = x h + k Vertical Asymptote: Occurs at x = 1 because if x = 1, the denominator, x 1, would be zero.

2 Page 2 1) y = x 1 2 First, note that h = 1 and k = 2 Recall that the simple rational a form is: y = x h + k Vertical Asymptote: Occurs at x = 1 because if x = 1, the denominator, x 1, would be zero. Horizontal Asymptote: Occurs at y = 2 because the lead term, x 1 zero. Hence, the function can approach y = 2, but will never reach it. Domain: All Real x 1. No value of x exists at a vertical asymptote., can never be Range: All Real y 2. No value of y exists at a horizontal asymptote in a simple rational function. Holes: None. End Behavior: Both ends of the function tend toward the horizontal asymptote, so: AA x, y 2 and AA x, y 2 Graphing: We know a lot about the function before even starting to graph it. Here are the steps I use to graph a simple rational function. Step 1. Graph the vertical and horizontal asymptotes (the dashed horizontal and vertical lines shown). Step 2. Pick some x-values and calculate the corresponding y-values. I like to pick a couple of xvalues to the left of the vertical asymptote (x = 1) and a couple of x-values to its right. So, let s try some. x y = x = 2 = = 2 2 = 4. 2 = 2 = = 1 2 = = 2 2 = = 3 2 = 0.33 Note that the dot at the intersection of the asymptotes has coordinates (h, k). You do NOT need to show this point on your test. Step 3. Draw a curve on each side of the vertical asymptote, through the points on that side, approaching both the horizontal and vertical asymptotes.

3 Page 3 2) y = 4x+19 x+3 First, let s convert this to a more useful form. y = 4x+19 x+3 constant + 4(x+3) = = +4(x+3) = x+3 x+3 + 4(x+3) = + 4 x+3 x+3 x+3 Since the denominator is (x + 3), I choose to make the numerator a constant plus a multiple of (x + 3). So, our converted equation becomes: y = 6 x Note that h = 3 and k = 4 in this form. Then, To find constant", I multiply 4 times the denominator: 4(x + 3) = 4x Then I ask what constant I need to make the numerator equal to what it was at the start: (4x + 19). That s how I come up with the value of constant. Vertical Asymptote: Horizontal Asymptote: Occurs at x = 3. Occurs at y = 4. Domain: All Real x 3. No value of x exists at a vertical asymptote. Range: All Real y 4. No value of y exists at the horizontal asymptote. Holes: None. End Behavior: Both ends of the function tend toward the horizontal asymptote, so: AA x, y 4 and AA x, y 4 Graphing: Step 1. Graph the vertical and horizontal asymptotes. Step 2. Pick some x-values on each side of the vertical asymptote and calculate the corresponding y-values. x y = 6 x = = = = = = = = = = = = 6.33 Step 3. Draw a curve on each side of the vertical asymptote, through the points on that side, approaching both the horizontal and vertical asymptotes.

4 Page 4 GENERAL RATIONAL FUNCTIONS Now, a couple of thoughts about general rational functions: y = P(x) The easiest way to graph a general rational function is to factor both the numerator and denominator and simplify by eliminating identical factors in the numerator and denominator. For example, if y = (x+3)(x+2) the (x + 2) in the numerator and denominator can be (x 1)(x+2) eliminated to obtain the function to be graphed: y = (x+3). You must remember, however, any (x 1) terms removed from the denominator, because this is where the holes in the graph will be located. In this example x = 2 will define the location of a hole. See more below. Vertical Asymptotes and Holes: Any root (also called a zero ) of the denominator of a rational function will produce either a vertical asymptote or a hole. Vertical Asymptote: If r is a root of the denominator and remains a root of the denominator after eliminating identical factors in the numerator and denominator, then x = r is a vertical asymptote of the function. Hole: If r is a root of the denominator and is eliminated from the denominator when eliminating identical factors in the numerator and denominator, then x = r defines the location of a hole in the function. Horizontal Asymptote: Horizontal asymptotes can be found by eliminating all terms of the polynomials in both the numerator and denominator except the ones with the single greatest exponent of all the terms. Then, If all terms are eliminated from the numerator, the horizontal asymptote occurs at y = 0. Example: y = x+3 x 2 x+6 Q(x) y = nothing x 2 has a horizontal asymptote at y = 0. Note that all terms in the numerator were eliminated because none of them had the greatest exponent in the rational function, which in this example is 2. If a term remains in both the numerator and denominator, the horizontal asymptote occurs at the reduced form of the remaining terms. Example: y = y = x2 +3 3x 2 x+6 y = 2x2 3x 2 = 2 3 has a horizontal asymptote at If all terms are eliminated from the denominator, the function does not have a horizontal asymptote. Example: y = x2 x+6 x 3 y = x2 nothing does not have a horizontal asymptote. Note that all terms in the denominator were eliminated because none of them had the greatest exponent in the rational function, which in this example is 2.

5 Page Domain: The domain is always all Real x except where there is a vertical asymptote or a hole. No function value is associated with x at either a vertical asymptote or a hole. Range: The range is a bit trickier. You will need to look at the graph to determine the range. You might think that no y-value would exist at a horizontal asymptote, like in simple rational functions. However, it is possible for a function to cross over its horizontal asymptote and then work its way back to the asymptote as x or x. Odd but true (see below, right). For oddities in the range of a function, check these out: y = x2 x 2 +3 y = 10x+ x 2 +6x+ End Behavior: Both ends of the function tend toward the horizontal asymptote if there is one. However, if there is not one, you must look at the graph to determine end behavior. Note that all of the functions in the sample test have a horizontal asymptote, but the function below does not: y = x3 x 2 6 In this function, AA x, y, AA x, y

6 Page 6 Now, let s do problems #3 6. 3) y = x2 x 2 Get the Roots: Numerator: x=0 Denominator: x=2 Vertical Asymptotes and Holes: 2 is a root of the denominator, so it must be either a vertical asymptote or a hole. Vertical Asymptote: Since this function cannot be simplified further, x = 2 is a vertical asymptote. Holes: None. Since the function cannot be simplified, no factors drop out of the denominator when the function is simplified. Horizontal Asymptote: Eliminate all terms of both polynomials except any with the single greatest exponent of all the terms. In this case: y = x2 x 2 y = x2 nothing so this function does not have a horizontal asymptote. Note that all terms in the denominator were eliminated because none of them had the greatest exponent in the rational function, which in this case is 2. Domain: All Real x except where there is a vertical asymptote or a hole. So, the domain is all Real x 2. Now, let s graph it in order to get a good look at the range and end behavior. We want points on each side of the vertical asymptote. x y = x2 x 2-1 ( 1) = 1 3 = = 0 2 = = 1 1 = = 9 1 = = 16 2 = = 2 3 = 8.33 Graphing: Step 1. Graph the vertical and horizontal asymptotes. Step 2. Pick some x-values on each side of the vertical asymptote and calculate the corresponding y-values. Step 3. Draw a curve on each side of the vertical asymptote, through the points on that side, approaching the vertical asymptote. (Graph on next Page)

7 Page Range: The range must be determined from the graph. It looks like a decent estimate of the range is: y 0 or y 8. End Behavior: In this function, As x, y, As x, y 4) y = x2 +9 x 2 2x 1 Only the denominator can be factored: y = x2 +9 (x+3)(x ) Get the Roots: Numerator: none Denominator: x= 3, x= Vertical Asymptotes and Holes: -3 and are roots of the denominator, so they must be either vertical asymptotes or holes. Vertical Asymptotes: Since this function cannot be simplified further, both x = 3 and x = are vertical asymptotes. Holes: None. Since the function cannot be simplified, no factors drop out of the denominator when the function is simplified. Horizontal Asymptote: Eliminate all terms of both polynomials except any with the single greatest exponent of all the terms. In this case: y = x2 y = 1 is a horizontal asymptote. Since a term remains in both the x2 numerator and denominator, the horizontal asymptote occurs at the reduced form of those remaining terms. Domain: All Real x except where there is a vertical asymptote or a hole. So, the domain is all Real x 3 or. Now, let s graph it in order to get a good look at the range and end behavior. We want points in all three regions carved out by the vertical asymptotes.

8 Page 8 Graphing: Step 1. Graph the vertical and horizontal asymptotes. Step 2. Pick x-values in all three regions created by the vertical asymptotes, and calculate the corresponding y-values. Notice that I selected 3 points to the left of the left vertical asymptote and 3 points to the right of the right vertical asymptote. In the middle region, it is a good idea to select a point close to each asymptote, and one in the middle of the region, which you can see I did. This gives you a good handle on the behavior of the middle region. In some cases, you may have to select more than 3 points. Also notice that I am using the factored form of the denominator to calculate my y-values. This is often easier than using the original unfactored polynomial. Step 3. Draw a curve in each of the three regions defined by the vertical asymptotes, through the points in that region, approaching both the horizontal and vertical asymptotes. x x y = (x + 3)(x ) -6 ( 6) ( 6 + 3)( 6 ) = 4 33 = ( ) ( + 3)( ) = = ( 4) ( 4 + 3)( 4 ) = 2 9 = ( 2) ( 2 + 3)( 2 ) = 13 = (1 + 3)(1 ) = = (4 + 3)(4 ) = 2 = (6 + 3)(6 ) = 4 9 = ( + 3)( ) = 8 20 = (8 + 3)(8 ) = 3 33 = 2.21 Range: The range must be determined from the graph. It looks like a decent estimate of the range is: y 0. or y 1. End Behavior: In this function, As x, y 1, As x, y 1

9 Page 9 ) y = x2 +x 6 x 2 x 2 Factor both the numerator and the denominator: y = (x 2)(x+3) (x+1)(x 2) Get the Roots: Numerator: x= 3, x=2 Denominator: x= 1, x=2 Since 2 is a root of both the numerator and the denominator, the function may be simplified as follows: y = (x 2)(x+3) (x+1)(x 2) = (x 2)(x+3) (x+1)(x 2) = x+3 x+1 Vertical Asymptotes and Holes: -1 and 2 are roots of the original denominator, so they must be either vertical asymptotes or holes. Vertical Asymptote: After simplification, this function still contains -1 as a root in the denominator. Therefore, x = 1 is a vertical asymptote of the function. Hole: 2 is a root of the denominator of the original function but is not a root of the denominator of the revised function. Therefore, this function has a hole at x = 2 Horizontal Asymptote: Eliminate all terms of both polynomials except any with the single greatest exponent of all the terms. In this case: y = x2 y = 1 is a horizontal asymptote. Since a term remains in both the x2 numerator and denominator, the horizontal asymptote occurs at the reduced form of those remaining terms. Domain: All Real x except where there is a vertical asymptote or a hole. So, the domain is all Real x 1 or 2. Now, let s graph it in order to get a good look at the range and end behavior. We want points on both sides of the vertical asymptote. (Graph on next Page)

10 Page 10 Graphing: Step 1. Graph the vertical and horizontal asymptotes. Step 2. Pick some x-values on each side of the vertical asymptote and calculate the corresponding y-values. x x + 3 y = x = 1 3 = = 0 2 = = 1 1 = = 3 1 = = 4 2 = = = 1.6 (a hole) Step 3. Draw a curve on each side of the vertical asymptote, through the points on that side, approaching both the horizontal and vertical asymptotes. Step 4: Draw an open circle at the point of any holes. Range: The range must be determined from the graph. It looks like the range excludes only the horizontal asymptote. So the range is: all Real y 1. See the hole at (2, 1. 66)! End Behavior: In this function, As x, y 1, As x, y 1

11 Page 11 6) y = 2x2 +x 3 x 2 1 Factor both the numerator and the denominator: y = (x 1)(2x+3) (x+1)(x 1) Get the Roots: Numerator: x= 1., x=1 Denominator: x= 1, x=1 Since 1 is a root of both the numerator and the denominator, the function may be simplified as follows: y = (x 1)(2x+3) (x+1)(x 1) = (x 1)(2x+3) (x+1)(x 1) = 2x+3 x+1 Vertical Asymptotes and Holes: -1 and 1 are roots of the original denominator, so they must be either vertical asymptotes or holes. Vertical Asymptote: After simplification, this function still contains -1 as a root in the denominator. Therefore, x = 1 is a vertical asymptote of the function. Hole: 1 is a root of the denominator of the original function but is not a root of the denominator of the revised function. Therefore, this function has a hole at x = 1 Horizontal Asymptote: Eliminate all terms of both polynomials except any with the single greatest exponent of all the terms. In this case: y = 2x2 y = 2 is a horizontal asymptote. Since a term remains in both the x2 numerator and denominator, the horizontal asymptote occurs at the reduced form of those remaining terms. Domain: All Real x except where there is a vertical asymptote or a hole. So, the domain is all Real x 1 or 1. Now, let s graph it in order to get a good look at the range and end behavior. We want points on both sides of the vertical asymptote. (Graph on next Page)

12 Page 12 Graphing: Step 1. Graph the vertical and horizontal asymptotes. Step 2. Pick some x-values on each side of the vertical asymptote and calculate the corresponding y-values. x x + 3 y = x + 1 2( 4) = 3 = 1.6 2( 3) = 3 2 = 1. 2( 2) = 1 1 = 1 2(0) = 3 1 = 3 2(1)+3 = = 2. (a hole) (2) = 3 = 2.33 Step 3. Draw a curve on each side of the vertical asymptote, through the points on that side, approaching both the horizontal and vertical asymptotes. Step 4: Draw an open circle at the point of any holes. Range: The range must be determined from the graph. It looks like the range excludes only the horizontal asymptote. So the range is: all Real y 1. See the hole at (1, 2. )! End Behavior: In this function, As x, y 1, As x, y 1 The next three pages provide printouts from the software I used to check my work for each function on the practice test. You do not need to look at these, but you may find them interesting.

13 Page 13 Problem #1 Problem #2

14 Page 14 Problem #3 Problem #4

15 Page 1 Problem # Problem #6

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