Graphing Rational Functions Study Notes

Fundamental Properties of Rational Functions

• A rational function is not continuous and has multiple branches ($n+1$ branches, where $n$ is the number of vertical asymptotes).
• Vertical Asymptote (VA): $x$ values where the denominator equals zero; represented by dotted/dashed lines.
• Horizontal Asymptote (HA): $y$ values the function approaches but never reaches.
• Holes: Occur when a factor is common to both the numerator and denominator and is divided out.
• Intercepts: $x$-intercepts occur at $y=0$; $y$-intercepts occur at $x=0$.
• Domain: All real numbers except vertical asymptotes and holes.
• Range: All real numbers except horizontal asymptotes.
• Parent Function: $y = \frac{1}{x}$ has a $VA$ at $x = 0$ and an $HA$ at $y = 0$.

Transformation Form: $y = \frac{a}{x-h} + k$

• Value of $a$: Sign dictates the graph's regions; numerical value indicates vertical stretch or compression.
• Variable $h$: Determines the vertical asymptote at $x = h$ (horizontal shift; opposite of the sign).
• Variable $k$: Determines the horizontal asymptote at $y = k$ (vertical shift; keeps the sign).
• Functions in this specific form do not contain holes.

Horizontal Asymptote (HA) Relationships

• BOBO: Bigger On Bottom, $y = 0$. (Degree of denominator > Degree of numerator).
• BOTN: Bigger On Top, None. (Degree of numerator > Degree of denominator; may have a slant/oblique asymptote found by division, though not in Math three).
• EATs DC: Exponents Are The Same, Divide Coefficients. (Degree of numerator = Degree of denominator, $y = \frac{a}{b}$).

Finding Holes and Vertical Asymptotes

• Step 1: Factor numerator and denominator completely.
• Step 2 (Holes): Identify shared factors. Write as a coordinate point $(x, y)$ by substituting the restricted $x$ value into the simplified equation.
• Step 3 (VAs): Identify remaining restrictions in the denominator and write as $x = ext{value}$.

Calculation Examples

• Example 1: $y = \frac{x^2+4x-5}{x^2+9x+20}$ simplified to $y = \frac{x-1}{x+4}$.   - Hole: $(-5, 6)$.   - $VA$: $x = -4$.   - $HA$: $y = 1$.   - Domain: $(- ext{infinity}, -5) imes (-5, -4) imes (-4, ext{infinity})$.
• Example 2: $y = \frac{x-3}{x^2-9}$ simplified to $y = \frac{1}{x+3}$.   - Hole: $(3, \frac{1}{6})$.   - $VA$: $x = -3$.   - $HA$: $y = 0$.   - Domain: $(- ext{infinity}, -3) imes (-3, 3) imes (3, ext{infinity})$.
• Example 3: $y = \frac{3x^2-12}{6x^2+30x+36}$ simplified to $y = \frac{x-2}{2 imes (x+3)}$.   - $HA$: $y = \frac{1}{2}$.   - Hole: $(-2, -2)$.   - $VA$: $x = -3$.

Questions & Discussion

• Prompt: Students were asked to solve question number four individually: $y = \frac{x^3-x^2-20x}{4x^2-4x-24}$.
• Response:   - Factored Form: $y = \frac{x imes (x-5) imes (x+4)}{4 imes (x-3) imes (x+2)}$.   - Vertical Asymptotes: $x = 3$ and $x = -2$.   - Horizontal Asymptote: None (Bigger on top: degree 3 vs degree 2).   - Holes: None (no shared factors).   - Domain: $(- ext{infinity}, -2) imes (-2, 3) imes (3, ext{infinity})$.