# 4_4 - Chapter 4 Polynomial and Rational Functions 4.4 1 3 5...

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Chapter 4 454 Polynomial and Rational Functions 4.4 Rational Functions II: Analyzing Graphs 1. False 2. (4, 2), (–4, –2) 3. in lowest terms 4. False 5. False 6. True In problems 7–44, we will use the terminology: R ( x ) = p ( x ) q ( x ) , where the degree of p ( x ) = n and the degree of q ( x ) = m . 7. R ( x ) = x + 1 x ( x + 4) p ( x ) = x + 1; q ( x ) = x ( x + 4) = x 2 + 4 x ; n = m = 2 Step 1: Domain: x x ≠ - 4, x 0 { } Step 2: (a) The x- intercept is the zero of p ( x ) : –1 (b) There is no y- intercept; R (0) is not defined, since q (0) = 0. Step 3: R ( - x ) = - x + 1 - x ( - x + 4) = - x + 1 x 2 - 4 x ; this is neither R ( x ) nor - R ( x ) , so there is no symmetry. Step 4: R ( x ) = x + 1 x ( x + 4) is in lowest terms. The vertical asymptotes are the zeros of q ( x ): x = - 4 and x = 0 . Step 5: Since n < m , the line y = 0 is the horizontal asymptote. Solve to find intersection points: x + 1 x ( x + 4) = 0 x + 1 = 0 x = - 1 R ( x ) intersects y = 0 at (–1, 0). Step 6:

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Section 4.4 Rational Functions II: Analyzing Graphs 455 Step 7: Graphing 8. R ( x ) = x ( x - 1)( x + 2) p ( x ) = x ; q ( x ) = ( x - 1)( x + 2) = x 2 + x - 2; n = 1; m = 2 Step 1: Domain: x x ≠ - 2, x 1 { } Step 2: (a) The x- intercept is the zero of p ( x ) : 0 (b) The y- intercept; R (0) = 0 Step 3: R ( - x ) = - x ( - x - 1)( - x + 2) = - x x 2 - x - 2 ; this is neither R ( x ) nor - R ( x ) , so there is no symmetry. Step 4: R ( x ) = x ( x - 1)( x + 2) is in lowest terms. The vertical asymptotes are the zeros of q ( x ): x = - 2 and x = 1. Step 5: Since n < m , the line y = 0 is the horizontal asymptote. Solve to find intersection points: x ( x - 1)( x + 2) = 0 x = 0 R ( x ) intersects y = 0 at (0, 0). Step 6:
Chapter 4 Polynomial and Rational Functions 456 Step 7: Graphing: 9. R ( x ) = 3 x + 3 2 x + 4 p ( x ) = 3 x + 3; q ( x ) = 2 x + 4; n = 1; m = 1 Step 1: Domain: x x ≠ - 2 { } Step 2: (a) The x- intercept is the zero of p ( x ) : –1 (b) The y- intercept is R (0) = 3(0) + 3 2(0) + 4 = 3 4 . Step 3: R ( - x ) = 3( - x ) + 3 2( - x ) + 4 = - 3 x + 3 - 2 x + 4 = 3 x - 3 2 x - 4 ; this is neither R ( x ) nor - R ( x ) , so there is no symmetry. Step 4: R ( x ) = 3 x + 3 2 x + 4 is in lowest terms. The vertical asymptote is the zero of q ( x ): x = - 2 Step 5: Since n = m , the line y = 3 2 is the horizontal asymptote. Solve to find intersection points: 3 x + 3 2 x + 4 = 3 2 2 3 x + 3 ( 29 = 3 2 x + 4 ( 29 6 x + 6 = 6 x + 4 0 2 R ( x ) does not intersect y = 3 2 . Step 6:

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Section 4.4 Rational Functions II: Analyzing Graphs 457 Step 7: Graphing: 10. R ( x ) = 2 x + 4 x - 1 p ( x ) = 2 x + 4; q ( x ) = x - 1; n = m = 1 Step 1: Domain: x x 1 { } Step 2: (a) The x- intercept is the zero of p ( x ) : –2 (b) The y- intercept is R (0) = 2(0) + 4 0 - 1 = 4 - 1 = - 4. Step 3: R ( - x ) = 2( - x ) + 4 ( - x ) - 1 = - 2 x + 4 - x - 1 = 2 x - 4 x + 1 ; this is neither R ( x ) nor - R ( x ) , so there is no symmetry. Step 4: R ( x ) = 2 x + 4 x - 1 is in lowest terms. The vertical asymptote is the zero of q ( x ): x = 1 Step 5: Since n = m , the line y = 2 is the horizontal asymptote. Solve to find intersection points: 2 x + 4 x - 1 = 2 2 x + 4 = 2 x - 1 ( 29 2 x + 4 = 2 x - 1 0 ≠ - 5 R ( x ) does not intersect y = 2 . Step 6:
Chapter 4 Polynomial and Rational Functions 458 Step 7: Graphing: 11. R ( x ) = 3 x 2 - 4 p ( x ) = 3; q ( x ) = x 2 - 4; n = 0; m = 2 Step 1: Domain: x x ≠ - 2, x 2 { } Step 2: (a) There is no x- intercept. (b) The y- intercept is R (0) = 3 0 2 - 4 = 3 - 4 =- 3 4 . Step 3: R ( - x ) = 3 ( - x ) 2 - 4 = 3 x 2 - 4 = R ( x ) ; R ( x ) is symmetric with respect to the y- axis.

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4_4 - Chapter 4 Polynomial and Rational Functions 4.4 1 3 5...

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