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: 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 2: 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 3: 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 4: Identify the vertical asymptote(s) of $f(x)=\frac{x^2-9}{x^2-4x+4}$.

Answer: $x=2$. Factor denominator: $(x-2)^2$ gives double root.

Flashcard 5: Identify the domain of $f(x)=\frac{1}{(x-2)(x+5)}$.

Answer: All real $x$ except $x=2$ and $x=-5$. Exclude zeros of denominator factors.

Flashcard 6: 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 7: Identify the vertical asymptote of $f(x)=\frac{x+2}{x-1}$.

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

Flashcard 8: Find the hole location for $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

Answer: Hole at $x=4$. Common factor $(x-4)$ creates hole.

Flashcard 9: 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 10: Identify the horizontal asymptote of $f(x)=\frac{x^2}{x^2-9}$.

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

Flashcard 11: What type of asymptote occurs when $\deg(p)=\deg(q)+1$ for $f(x)=\frac{p(x)}{q(x)}$?

Answer: A slant (oblique) asymptote from polynomial division. Numerator one degree higher creates slant asymptote.

Flashcard 12: Identify the horizontal asymptote of $f(x)=\frac{x+2}{x-1}$.

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

Flashcard 13: 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 14: What is the general form of a rational function $f(x)$?

Answer: $f(x)=\frac{p(x)}{q(x)}$ where $p(x),q(x)$ are polynomials and $q(x)\neq 0$. Ratio of polynomials with non-zero denominator.

Flashcard 15: Identify the $x$-intercept(s) of $f(x)=\frac{x^2}{x^2-9}$.

Answer: $x=0$. Numerator $x^2$ is zero when $x=0$.

Flashcard 16: 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 17: What is the horizontal asymptote when $\deg(p)=\deg(q)$ for $f(x)=\frac{p(x)}{q(x)}$?

Answer: $y=\frac{\text{leading coeff of }p}{\text{leading coeff of }q}$. Equal degrees give ratio of leading coefficients.

Flashcard 18: Identify the end behavior of $f(x)=\frac{-2x^5+1}{x^4-7}$ as $x\to\pm\infty$.

Answer: No horizontal asymptote; $f(x)\sim -2x$. Numerator degree exceeds denominator by 1.

Flashcard 19: Identify the vertical asymptote(s) of $f(x)=\frac{1}{x-3}$.

Answer: $x=3$. Denominator is zero when $x-3=0$.

Flashcard 20: 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 21: 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 22: 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 23: 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 24: 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 25: Identify the $x$-intercept(s) of $f(x)=\frac{x^2-9}{x^2-4x+4}$.

Answer: $x=-3$ and $x=3$. Factor numerator: $(x-3)(x+3)$ gives zeros.

Flashcard 26: Identify the vertical asymptote(s) of $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

Answer: $x=1$. After canceling $(x-4)$, only $x-1=0$ remains.

Flashcard 27: 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 28: Identify the $x$-intercept(s) of $f(x)=\frac{(x-2)(x+1)}{x-5}$.

Answer: $x=2$ and $x=-1$. Set numerator factors equal to zero.

Flashcard 29: What is the domain rule for $f(x)=\frac{p(x)}{q(x)}$ before simplifying?

Answer: All real $x$ except where $q(x)=0$. Exclude values where denominator equals zero.

Flashcard 30: Identify whether $f(x)=\frac{x-1}{x-1}$ has a vertical asymptote or a hole at $x=1$.

Answer: A hole at $x=1$. Common factor $(x-1)$ cancels completely.

Flashcard 31: 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 32: 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 33: 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 34: What is the end behavior idea for a rational function $f(x)$?

Answer: As $x\to\pm\infty$, $f(x)$ approaches its asymptote (if any). Function approaches asymptote as x approaches infinity.

Flashcard 35: 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 36: 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 37: Identify the horizontal asymptote of $f(x)=\frac{5}{x^2+1}$.

Answer: $y=0$. Degree of numerator less than denominator.

Flashcard 38: 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 39: 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 40: 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 41: 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 42: 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 43: Identify the $x$-intercept(s) of $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

Answer: $x=-2$. From remaining numerator factor $(x+2)$.

Flashcard 44: Identify the $x$-intercept of $f(x)=\frac{x+2}{x-1}$.

Answer: $x=-2$. Set numerator equal to zero: $x+2=0$.

Flashcard 45: What is the sign-based test to determine the side behavior near a vertical asymptote $x=a$?

Answer: Check the sign of $f(x)$ on each side using factor signs. Test sign changes across vertical asymptotes.

Flashcard 46: Find the hole point for $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

Answer: Hole at $\left(4,\frac{6}{3}\right)=\left(4,2\right)$. Substitute $x=4$ into simplified function.

Flashcard 47: 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 48: Identify the vertical asymptote of $f(x)=\frac{x^2+1}{x}$.

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

Flashcard 49: Identify the horizontal asymptote of $f(x)=\frac{(x+1)^2}{(x-3)^3}$.

Answer: $y=0$. Numerator degree less than denominator degree.

Flashcard 50: Identify the vertical asymptote(s) of $f(x)=\frac{(x-2)(x+1)}{(x-5)(x+3)}$.

Answer: $x=5$ and $x=-3$. Set denominator factors equal to zero.

Flashcard 51: 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 52: 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 53: What is the horizontal asymptote of $f(x)=\frac{-7x^4+1}{2x^4-3x}$?

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