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

2. OBJECTIVE 1

4. 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

7. 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 = x
x x
implies y = x + 3 is oblique horizontal asymptote

10. 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 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

13. Step 1: f(x) = x2 - 9 = (x – 3)(x + 3)
Domain: x ≠ -6, 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 – 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

16. 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,

17. OBJECTIVE 2

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