Algebra 2 Flashcards: Graph Rational Functions And Identify Asymptotes

What this deck covers

This deck focuses on Graph Rational Functions And Identify Asymptotes, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the $x$-intercept rule for a rational function in factored form?

Answer: Zeros of the numerator that are not canceled by the denominator. Numerator zeros give x-intercepts unless cancelled out.

Flashcard 2: What is the vertical asymptote rule for $f(x)=\frac{p(x)}{q(x)}$ in factored form?

Answer: Zeros of $q(x)$ that do not cancel with $p(x)$. Denominator zeros cause vertical asymptotes unless cancelled.

Flashcard 3: Identify the $y$-intercept of $f(x)=\frac{x+2}{x-1}$.

Answer: $f(0)=-2$, so $\left(0,-2\right)$. Substitute $x=0$: $\frac{0+2}{0-1}=-2$.

Flashcard 4: What is the hole (removable discontinuity) rule for a rational function in factored form?

Answer: A common factor cancels; the hole is at that factor's zero. Common factors create holes at their zeros.

Flashcard 5: What is the $y$-intercept of $f(x)$, if it exists?

Answer: $f(0)$, provided $q(0)\neq 0$. Substitute $x=0$ if denominator is non-zero.

Flashcard 6: What is the horizontal asymptote when $\deg(p)<\deg(q)$ for $f(x)=\frac{p(x)}{q(x)}$?

Answer: $y=0$. Lower degree numerator approaches zero at infinity.

Flashcard 7: Identify the horizontal asymptote of $f(x)=\frac{x^2}{x^2-9}$.

Answer: $y=1$. Equal degrees: $\frac{1}{1}=1$.

Flashcard 8: Identify the vertical asymptote(s) of $f(x)=\frac{x^2}{x^2-9}$.

Answer: $x=-3$ and $x=3$. Factor denominator: $x^2-9=(x-3)(x+3)$.

Flashcard 9: What is the key difference between a hole and a vertical asymptote at $x=a$?

Answer: Hole: factor cancels; VA: factor remains in denominator. Cancellation creates holes; non-cancellation creates VAs.

Flashcard 10: Identify whether the graph crosses or touches the $x$-axis at $x=-1$ for $f(x)=\frac{(x+1)^2}{x-3}$.

Answer: Touches (even multiplicity at $x=-1$). Even multiplicity at x-intercept means touching.

Flashcard 11: Identify the vertical asymptote(s) of $f(x)=\frac{(x+1)^2}{(x-3)^3}$.

Answer: $x=3$. Denominator $(x-3)^3$ has odd multiplicity 3.

Flashcard 12: Identify the behavior near $x=2$ for $f(x)=\frac{1}{x-2}$.

Answer: As $x\to 2^{-}$, $f(x)\to -\infty$; as $x\to 2^{+}$, $f(x)\to +\infty$. Odd multiplicity creates opposite infinities on each side.

Flashcard 13: Identify the behavior near $x=2$ for $f(x)=\frac{1}{(x-2)^2}$.

Answer: As $x\to 2^{\pm}$, $f(x)\to +\infty$. Even multiplicity makes both sides approach same infinity.

Flashcard 14: What is the multiplicity effect near a vertical asymptote $(x-a)^k$ in the denominator?

Answer: Odd $k$: opposite infinities; even $k$: same infinity on both sides. Multiplicity affects sign behavior near asymptotes.

Flashcard 15: What is the multiplicity effect on $x$-intercepts for a factor $(x-a)^k$ in the numerator?

Answer: Odd $k$: crosses; even $k$: touches and turns at $x=a$. Multiplicity determines crossing vs touching behavior.

Flashcard 16: What is the correct order of steps to graph $f(x)=\frac{p(x)}{q(x)}$ from a factored form?

Answer: Find holes, VAs, HAs/slant, intercepts, then sketch behavior. Systematic approach ensures all features are identified.

Flashcard 17: Find the hole point for $f(x)=\frac{x^2-4}{x-2}$.

Answer: Hole at $\left(2,4\right)$. Substitute $x=2$ into simplified form $x+2$.

Flashcard 18: Identify whether $f(x)=\frac{x^2-4}{x-2}$ has a hole or vertical asymptote at $x=2$.

Answer: A hole at $x=2$. Factor $(x-2)$ cancels from numerator $(x^2-4)$.

Flashcard 19: What is the slant asymptote of $f(x)=\frac{x^2-4}{x-2}$?

Answer: $y=x+2$. Factor and cancel: $\frac{x^2-4}{x-2}=x+2$ after division.

Flashcard 20: Identify the vertical asymptote of $f(x)=\frac{x^2+1}{x}$.

Answer: $x=0$. Denominator is zero when $x=0$.

Flashcard 21: What is the slant asymptote of $f(x)=\frac{x^2+1}{x}$?

Answer: $y=x$. Polynomial division: $\frac{x^2+1}{x}=x+\frac{1}{x}$.

Flashcard 22: Identify the simplified function for $f(x)=\frac{x^2-1}{x-1}$ for $x\neq 1$.

Answer: $f(x)=x+1$ for $x\neq 1$. After canceling $(x-1)$: $\frac{x^2-1}{x-1}=x+1$.

Flashcard 23: Find the hole point for $f(x)=\frac{x^2-1}{x-1}$.

Answer: Hole at $\left(1,2\right)$. Substitute $x=1$ into simplified form $x+1$.

Flashcard 24: Identify whether $f(x)=\frac{x^2-1}{x-1}$ has a hole or vertical asymptote at $x=1$.

Answer: A hole at $x=1$. Factor $(x-1)$ cancels from numerator and denominator.

Flashcard 25: Identify the horizontal asymptote of $f(x)=\frac{x^2-9}{x^2-4x+4}$.

Answer: $y=1$. Equal degrees: $\frac{1}{1}=1$.

Flashcard 26: Identify the horizontal asymptote of $f(x)=\frac{2x^3}{-4x^3+7}$.

Answer: $y=-\frac{1}{2}$. Leading coefficient ratio: $\frac{2}{-4}=-\frac{1}{2}$.

Flashcard 27: What is the horizontal asymptote of $f(x)=\frac{-7x^4+1}{2x^4-3x}$?

Answer: $y=-\frac{7}{2}$. Equal degrees: $\frac{-7}{2}=-\frac{7}{2}$.

Flashcard 28: What is the end behavior of $f(x)=\frac{4x^3-x}{2x^3+7}$ as $x\to\pm\infty$?

Answer: $f(x)\to 2$. Equal degrees: $\frac{4}{2}=2$.

Flashcard 29: What is the end behavior of $f(x)=\frac{3x^2+1}{x^3-5}$ as $x\to\pm\infty$?

Answer: $f(x)\to 0$. Numerator degree less than denominator degree.

Flashcard 30: Identify the vertical asymptote of $f(x)=\frac{x+2}{x-1}$.

Answer: $x=1$. Denominator is zero when $x-1=0$.