Graph of a Rational Function

INTRODUCTION AND IMPORTANCE OF GRAPHICAL REPRESENTATION

A rational function is given in the form:
f(x) = P(x) / Q(x),
where P(x) and Q(x) are polynomials, and Q(x) ≠ 0. Its graph is not a continuous curve like that of polynomials but contains characteristic breaks, asymptotes, holes, and other special points. The purpose of the graph is to show the behavior of the function around its zeros and points where it is not defined.

STEPS FOR GRAPHING A RATIONAL FUNCTION

1. DETERMINE THE DOMAIN (Df)
   The domain Df includes all real numbers except those where Q(x) = 0. These points are excluded because division by zero is not allowed.
   Example: f(x) = (x² – 1) / (x – 2) → Q(x) = x – 2 → Df = ℝ \ {2} (all real numbers except 2)
2. SIMPLIFY THE FUNCTION (IF POSSIBLE)
   Sometimes the numerator and denominator can be factored and common factors canceled. Remember that points corresponding to canceled factors from the denominator still represent holes and are not in the domain.
   Example: f(x) = (x² – 1) / (x – 2) = [(x – 1)(x + 1)] / (x – 2). (No common factors to cancel here).
3. DETERMINE THE ZEROS (X-INTERCEPTS) OF THE FUNCTION
   Zeros are the solutions to the equation P(x) = 0, provided that Q(x) ≠ 0 at these points. These points correspond to the intersections with the x-axis.
   In the above example: P(x) = (x – 1)(x + 1) → Zeros: x = –1, x = 1.
4. DETERMINE THE Y-INTERCEPT
   Substitute x = 0, if 0 ∈ Df:
   Example: f(0) = (0² – 1) / (0 – 2) = –1 / (–2) = 0.5. The y-intercept is (0, 0.5).
5. FIND VERTICAL ASYMPTOTES
   These occur at x-values where Q(x) = 0 after simplification (i.e., the factor in the denominator did not cancel with one in the numerator), and P(x) ≠ 0 at these x-values. At these points, the function tends towards ±∞, and the graph approaches a vertical line.
   Example: For f(x) = (x² – 1)/(x – 2), Q(x) = x – 2 → Vertical asymptote at x = 2.
6. FIND HORIZONTAL OR SLANT (OBLIQUE) ASYMPTOTES
   These are determined by comparing the degrees of the numerator P(x) and the denominator Q(x):
7. • If deg P < deg Q → Horizontal asymptote y = 0.
   • If deg P = deg Q → Horizontal asymptote y = aₙ/bₙ (ratio of leading coefficients).
   • If deg P > deg Q:
   • • If deg P = deg Q + 1 → Slant (oblique) asymptote, found by polynomial long division.
     • If deg P > deg Q + 1 → No horizontal or slant asymptote (behavior is polynomial-like).
       Example: f(x) = (2x² + 1) / (x² – 3) → deg P = deg Q = 2. Horizontal asymptote: y = 2/1 = 2.
7. IDENTIFY HOLES IN THE GRAPH (REMOVABLE DISCONTINUITIES)
   If a factor (x – c) appears in both the numerator and the denominator and can be canceled, then there is a hole in the graph at x = c. The function is not defined at x=c, but the graph approaches the point that would exist if the function were defined there.
   Example: f(x) = [(x – 1)(x + 2)] / (x – 1). This simplifies to f(x) = x + 2, for x ≠ 1.
   → The graph is a straight line with a hole at x = 1. To find the y-coordinate of the hole, plug x=1 into the simplified expression: y = 1 + 2 = 3. Hole at (1, 3).
8. ANALYZE BEHAVIOR AROUND ASYMPTOTES AND KEY POINTS
9. • To the left and right of a vertical asymptote, the function often tends towards +∞ or -∞. Test points to determine the direction.
   • For large values of |x|, the function's graph "hugs" or approaches the horizontal or slant asymptote.
   • The sign of the function can be determined in intervals between zeros and vertical asymptotes.

EXAMPLE OF A COMPLETE ANALYSIS

f(x) = (x² – 4) / (x – 1)

1. Simplify: f(x) = [(x – 2)(x + 2)] / (x – 1) (No common factors)
2. Domain (Df): ℝ \ {1}
3. Zeros: P(x) = (x – 2)(x + 2) = 0 → x = –2, x = 2.
4. Y-intercept: f(0) = (0² – 4) / (0 – 1) = –4 / –1 = 4. Point (0, 4).
5. Vertical Asymptote: Q(x) = x – 1 = 0 → x = 1.
6. Horizontal/Slant Asymptote: Degree of P(x) is 2, degree of Q(x) is 1. Since deg P = deg Q + 1, there is a slant asymptote. Perform polynomial long division of (x² – 4) by (x – 1).
   The quotient is x + 1 and the remainder is -3.
   So, f(x) = x + 1 - 3/(x – 1). The slant asymptote is y = x + 1.

CONCLUSION

The graph of a rational function contains important information about zeros, asymptotes, break points (discontinuities), and the behavior of the function for large |x|. A careful analysis of the numerator and denominator allows for a complete graphical understanding of the function. Special attention should be paid to asymptotes and holes, as these determine the main features of the graph's course.