Rational Functions, End Behavior, and Polynomial Characteristics

Vocabulary flashcards covering graphing rational functions (domain, asymptotes, holes, intercepts) and polynomial characteristics (end behavior, sign rules, and Pascal's triangle coefficients).

Domain (Rational Functions)

Determined from the denominator, including crossed-out factors (holes), with the sign flipped.

Zeros (Rational Functions)

Determined from the numerator without the crossed-out factors (holes), with the sign flipped.

Holes

Values found where factors are crossed out in both numerator and denominator, with the sign flipped (for example, at $x = 3$).

Vertical Asymptote

Found from the denominator of the simplified version without the hole, with the sign flipped.

Horizontal Asymptote (Bottom Higher Degree)

When the highest degree is in the denominator (bottom higher), the horizontal asymptote is $y = 0$.

Horizontal Asymptote (Equal Degrees)

When the degrees of the numerator and denominator are equal, the horizontal asymptote is found by dividing the highest-degree terms from the original version.

Horizontal Asymptote (Top Higher Degree)

When the degree of the numerator (top) is higher than the denominator, $HA = y = \text{NONE}$.

Slant Asymptote

Occurs when the degree of the numerator (top) is higher than the denominator by exactly $1$.

X-intercept (Rational Functions)

Equal to the zeros of the function, written as coordinate points (for example, $(3, 0)$).

Y-intercept (Rational Functions)

The point found by substituting/plugging in $0$ for $x$ into the original version of the function.

End Behaviors

Determined by looking at the term with the highest degree number (such as $3x^3$) and checking whether the leading coefficient is positive or negative, and whether the degree is even or odd.

End Behavior: Even Degree, Positive Coefficient

Both ends of the function point upward in the positive direction.

End Behavior: Even Degree, Negative Coefficient

Both ends of the function point downward in the negative direction.

End Behavior: Odd Degree, Positive Coefficient

The left end points downward (negative) and the right end points upward (positive).

End Behavior: Odd Degree, Negative Coefficient

The left end points upward (positive) and the right end points downward (negative).

Sign Behavior at Roots (Even vs. Odd Multiplicity)

Moving from the right: an even multiplicity keeps the same sign, while an odd multiplicity switches sign.

Pascal's Triangle Row 7 Coefficients

The set of binomial coefficients consisting of $1$, $7$, $21$, $35$, $35$, $21$, $7$, and $1$.