The Graph of a Rational Function Section 5.3 The Graph of a Rational Function

OBJECTIVE 1

Step 1: f(x) = x2 – 4 = (x – 2)(x + 2 ) Domain: x ≠ -4 and x ≠ 1 Range: All Reals Step 2: Can not simplify further Step 3: x-intercepts (-2, 0) and (2, 0) y-intercept: (0, 1) Step 4: f(-x) = x2 – 4 / x2 - 3x – 4 Not symmetry to y-axis or origin Step 5: Vertical asymptotes at x = -4 and x = 1 Step 6: x2 = 1 implies y = 1 is horizontal asymptote x2

Step 1: f(x) = x2 + 3x +2 = (x + 1)(x + 2) Domain: x ≠ 0 Range: All y ≥ 5.83 and y ≤ .17 Step 2: Can not simplify further Step 3: x-intercepts (-1, 0) and (-2, 0) y-intercept: none Step 4: f(-x) = x2 - 3x + 2 Not symmetry to y-axis or origin -x Step 5: Vertical asymptotes at x = 0 Step 6: f(x) = x2 + 3x + 2 = x + 3 + 2 x x implies y = x + 3 is oblique horizontal asymptote

Step 1: f(x) = x2 - x + 12 = (x2 - x + 12) Domain: x ≠ -1, 1 Range: All y > 1 and y ≤ -12 Step 2: Can not simplify further Step 3: x-intercepts: none y-intercept: (0, -12) Step 4: f(-x) = x2 + x + 12 Not symmetry to y-axis or origin x2 - 1 Step 5: Vertical asymptotes at x = 1 and x = -1 Step 6: x2 implies y = 1 is horizontal asymptote x2

Step 1: f(x) = x2 - 9 = (x – 3)(x + 3) Domain: x ≠ -6, -3 Range: { y | y ≠ 1} Step 2: f(x) = x – 3 x + 6 Step 3: x-intercepts: (3, 0) y-intercept: (0, -1/2) Step 4: f(-x) = x2 – 9 Not symmetry to y-axis or origin x2 – 9x + 18 Step 5: Vertical asymptotes at x = -6 and hole at x = -3 Step 6: x2 implies y = 1 is horizontal asymptote x2

Since the vertical asymptotes are x = -5 and x = 2 the denominator will have factors of (x + 5) and (x – 2). As x  -5-, R(x) approaches +∞ and as x  -5+, R(x) approaches -∞ implies (x + 5) is a factor of odd multiplicity. As x  2-, R(x) approaches -∞ and as x  2+, R(x) approaches -∞ implies (x - 2) is a factor of even multiplicity. So the denominator could be (x + 5)(x – 2)2. The x-intercepts determine the numerator. Since the x-intercepts are x = -2 with even multiplicity (touches x-axis) and x = 5 with odd multiplicity (crosses x-axis), the numerator will have (x + 2)2(x – 5) as part of the numerator. Since y = 2 is a horizontal asymptote the quotient of the leading coefficients of the numerator and denominator must be 2. Therefore we get,

OBJECTIVE 2

(a) C = 2 (πr2) (0.05) + (2πrh) (0.02) Volume = 500 = πr2h  h = 500/πr2 C = .10πr2 + .04πr(500/πr2) C = .10πr2 + 20/r C = .10πr3 + 20 r (b) (c) C(3.17) = 9.47