How to Sketch A Rational Function Without A Calculator

Introduction

A rational function is a fraction where both the numerator and denominator are polynomials. The general form is:

Sketching these functions involves identifying key features like asymptotes, intercepts, and behavior at critical points. While graphing calculators can provide quick visualizations, understanding these techniques helps you analyze functions deeply.

Basic Steps to Sketch a Rational Function

1. Identify the numerator and denominator polynomials.
2. Find vertical asymptotes by solving Q(x) = 0.
3. Find horizontal asymptotes by comparing degrees of P(x) and Q(x).
4. Plot x-intercepts by solving P(x) = 0.
5. Plot y-intercept by evaluating f(0).
6. Determine the behavior of the function near asymptotes and intercepts.
7. Sketch the curve using this information.

Identify the Components

Start by clearly identifying the numerator (P(x)) and denominator (Q(x)) polynomials. For example, consider the function:

Here, P(x) = x² - 4 and Q(x) = x² - 1.

Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (Q(x) = 0) and the numerator is not zero at the same point. Solve Q(x) = 0:

Check if the numerator is zero at these points:

Both points are vertical asymptotes.

Find Horizontal Asymptotes

Horizontal asymptotes are found by comparing the degrees of P(x) and Q(x):

• If degree of P(x) < degree of Q(x): y = 0
• If degree of P(x) = degree of Q(x): y = leading coefficient ratio
• If degree of P(x) > degree of Q(x): no horizontal asymptote (may have oblique asymptote)

In our example, both degrees are equal (2):

The horizontal asymptote is y = 1.

Plot Intercepts

Find x-intercepts by solving P(x) = 0:

Find y-intercept by evaluating f(0):

The intercepts are at (2,0), (-2,0), and (0,4).

Sketch the Curve

Using the information gathered:

1. Draw vertical asymptotes at x = 1 and x = -1.
2. Draw horizontal asymptote at y = 1.
3. Plot intercepts at (2,0), (-2,0), and (0,4).
4. Determine the behavior of the curve:
   • Approaches y = 1 as x → ±∞
   • Approaches ±∞ near x = ±1
5. Connect the points smoothly, showing the curve dipping below y = 1 between x = -2 and x = -1, and rising above y = 1 between x = -1 and x = 2.

Worked Example

Let's sketch f(x) = (x² - 4)/(x² - 1).

1. Identify components: P(x) = x² - 4, Q(x) = x² - 1.
2. Vertical asymptotes: x = ±1.
3. Horizontal asymptote: y = 1.
4. Intercepts: (2,0), (-2,0), (0,4).
5. Behavior:
   • As x → ±∞, f(x) → 1.
   • As x → 1⁻, f(x) → -∞; as x → 1⁺, f(x) → +∞.
   • As x → -1⁻, f(x) → +∞; as x → -1⁺, f(x) → -∞.

The resulting graph will show the curve approaching y = 1 from below between x = -2 and x = -1, and from above between x = -1 and x = 2, with vertical asymptotes at x = ±1 and intercepts at the calculated points.