4.1 Representation of Rational Functions

Rational functions are the ratio of two polynomials, with a domain excluding values that make the denominator zero. They're key to understanding complex mathematical relationships and have wide-ranging applications in various fields.

Graphing rational functions involves finding zeros, poles, and asymptotes. By analyzing these elements, we can sketch accurate graphs, revealing the function's behavior and helping us solve real-world problems involving rates and proportions.

Rational Functions

Definition of rational functions

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• Ratio of two polynomials $f(x) = \frac{P(x)}{Q(x)}$
  • $P(x)$ represents the numerator polynomial
  • $Q(x)$ represents the denominator polynomial, where $Q(x) \neq 0$
• Domain consists of all real numbers except values that make the denominator equal to zero
  • Find the domain by setting $Q(x) = 0$ and solving for $x$
  • Solutions are the excluded values from the domain (vertical asymptotes)

Zeros and poles identification

• Zeros (roots) are $x$-values where the numerator $P(x) = 0$
  • Find zeros by setting $P(x) = 0$ and solving for $x$
  • Zeros represent x-intercepts on the graph ($y = 0$)
• Poles are $x$-values where the denominator $Q(x) = 0$
  • Poles are excluded from the domain (vertical asymptotes)
  • Multiplicity of a pole is the number of times the factor appears in $Q(x)$
    • Higher multiplicity leads to more severe asymptotic behavior near the pole

Asymptotes of rational functions

• Horizontal asymptotes depend on the degrees of $P(x)$ and $Q(x)$
  • If $degree(P(x)) < degree(Q(x))$, the horizontal asymptote is $y = 0$
  • If $degree(P(x)) = degree(Q(x))$, the horizontal asymptote is $y = \frac{a_n}{b_m}$
    • $a_n$ and $b_m$ are leading coefficients of $P(x)$ and $Q(x)$, respectively
  • If $degree(P(x)) = degree(Q(x)) + 1$, there is an oblique (slant) asymptote
    • No horizontal asymptote in this case
• Vertical asymptotes occur at the poles ($x$-values where $Q(x) = 0$)
  • Vertical asymptote is a line $x = a$, where $a$ is a pole

Graphing rational functions

1. Find the domain and identify holes (removable discontinuities)
2. Locate zeros (x-intercepts) by setting $P(x) = 0$ and solving for $x$
3. Determine the y-intercept by evaluating $f(0)$
4. Find vertical asymptotes by setting $Q(x) = 0$ and solving for $x$
5. Determine horizontal or oblique asymptote based on the degrees of $P(x)$ and $Q(x)$
6. Analyze function behavior near zeros, poles, and asymptotes
   • Examine the sign of the function on either side of these key points
7. Plot key points (zeros, y-intercept, holes) and sketch the curve
   • Consider asymptotic behavior and sign changes in each interval
   • Connect the points smoothly, following the trends established by the asymptotes and sign analysis