Graph Rational Functions and Identify Asymptotes - Algebra 2
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What is the $x$-intercept rule for a rational function in factored form?
What is the $x$-intercept rule for a rational function in factored form?
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Zeros of the numerator that are not canceled by the denominator. Numerator zeros give x-intercepts unless cancelled out.
Zeros of the numerator that are not canceled by the denominator. Numerator zeros give x-intercepts unless cancelled out.
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What is the vertical asymptote rule for $f(x)=\frac{p(x)}{q(x)}$ in factored form?
What is the vertical asymptote rule for $f(x)=\frac{p(x)}{q(x)}$ in factored form?
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Zeros of $q(x)$ that do not cancel with $p(x)$. Denominator zeros cause vertical asymptotes unless cancelled.
Zeros of $q(x)$ that do not cancel with $p(x)$. Denominator zeros cause vertical asymptotes unless cancelled.
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Identify the $y$-intercept of $f(x)=\frac{x+2}{x-1}$.
Identify the $y$-intercept of $f(x)=\frac{x+2}{x-1}$.
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$f(0)=-2$, so $\left(0,-2\right)$. Substitute $x=0$: $\frac{0+2}{0-1}=-2$.
$f(0)=-2$, so $\left(0,-2\right)$. Substitute $x=0$: $\frac{0+2}{0-1}=-2$.
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What is the hole (removable discontinuity) rule for a rational function in factored form?
What is the hole (removable discontinuity) rule for a rational function in factored form?
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A common factor cancels; the hole is at that factor's zero. Common factors create holes at their zeros.
A common factor cancels; the hole is at that factor's zero. Common factors create holes at their zeros.
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What is the $y$-intercept of $f(x)$, if it exists?
What is the $y$-intercept of $f(x)$, if it exists?
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$f(0)$, provided $q(0)\neq 0$. Substitute $x=0$ if denominator is non-zero.
$f(0)$, provided $q(0)\neq 0$. Substitute $x=0$ if denominator is non-zero.
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What is the horizontal asymptote when $\deg(p)<\deg(q)$ for $f(x)=\frac{p(x)}{q(x)}$?
What is the horizontal asymptote when $\deg(p)<\deg(q)$ for $f(x)=\frac{p(x)}{q(x)}$?
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$y=0$. Lower degree numerator approaches zero at infinity.
$y=0$. Lower degree numerator approaches zero at infinity.
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Identify the horizontal asymptote of $f(x)=\frac{x^2}{x^2-9}$.
Identify the horizontal asymptote of $f(x)=\frac{x^2}{x^2-9}$.
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$y=1$. Equal degrees: $\frac{1}{1}=1$.
$y=1$. Equal degrees: $\frac{1}{1}=1$.
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Identify the vertical asymptote(s) of $f(x)=\frac{x^2}{x^2-9}$.
Identify the vertical asymptote(s) of $f(x)=\frac{x^2}{x^2-9}$.
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$x=-3$ and $x=3$. Factor denominator: $x^2-9=(x-3)(x+3)$.
$x=-3$ and $x=3$. Factor denominator: $x^2-9=(x-3)(x+3)$.
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What is the key difference between a hole and a vertical asymptote at $x=a$?
What is the key difference between a hole and a vertical asymptote at $x=a$?
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Hole: factor cancels; VA: factor remains in denominator. Cancellation creates holes; non-cancellation creates VAs.
Hole: factor cancels; VA: factor remains in denominator. Cancellation creates holes; non-cancellation creates VAs.
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Identify whether the graph crosses or touches the $x$-axis at $x=-1$ for $f(x)=\frac{(x+1)^2}{x-3}$.
Identify whether the graph crosses or touches the $x$-axis at $x=-1$ for $f(x)=\frac{(x+1)^2}{x-3}$.
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Touches (even multiplicity at $x=-1$). Even multiplicity at x-intercept means touching.
Touches (even multiplicity at $x=-1$). Even multiplicity at x-intercept means touching.
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Identify the vertical asymptote(s) of $f(x)=\frac{(x+1)^2}{(x-3)^3}$.
Identify the vertical asymptote(s) of $f(x)=\frac{(x+1)^2}{(x-3)^3}$.
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$x=3$. Denominator $(x-3)^3$ has odd multiplicity 3.
$x=3$. Denominator $(x-3)^3$ has odd multiplicity 3.
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Identify the behavior near $x=2$ for $f(x)=\frac{1}{x-2}$.
Identify the behavior near $x=2$ for $f(x)=\frac{1}{x-2}$.
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As $x\to 2^{-}$, $f(x)\to -\infty$; as $x\to 2^{+}$, $f(x)\to +\infty$. Odd multiplicity creates opposite infinities on each side.
As $x\to 2^{-}$, $f(x)\to -\infty$; as $x\to 2^{+}$, $f(x)\to +\infty$. Odd multiplicity creates opposite infinities on each side.
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Identify the behavior near $x=2$ for $f(x)=\frac{1}{(x-2)^2}$.
Identify the behavior near $x=2$ for $f(x)=\frac{1}{(x-2)^2}$.
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As $x\to 2^{\pm}$, $f(x)\to +\infty$. Even multiplicity makes both sides approach same infinity.
As $x\to 2^{\pm}$, $f(x)\to +\infty$. Even multiplicity makes both sides approach same infinity.
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What is the multiplicity effect near a vertical asymptote $(x-a)^k$ in the denominator?
What is the multiplicity effect near a vertical asymptote $(x-a)^k$ in the denominator?
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Odd $k$: opposite infinities; even $k$: same infinity on both sides. Multiplicity affects sign behavior near asymptotes.
Odd $k$: opposite infinities; even $k$: same infinity on both sides. Multiplicity affects sign behavior near asymptotes.
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What is the multiplicity effect on $x$-intercepts for a factor $(x-a)^k$ in the numerator?
What is the multiplicity effect on $x$-intercepts for a factor $(x-a)^k$ in the numerator?
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Odd $k$: crosses; even $k$: touches and turns at $x=a$. Multiplicity determines crossing vs touching behavior.
Odd $k$: crosses; even $k$: touches and turns at $x=a$. Multiplicity determines crossing vs touching behavior.
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What is the correct order of steps to graph $f(x)=\frac{p(x)}{q(x)}$ from a factored form?
What is the correct order of steps to graph $f(x)=\frac{p(x)}{q(x)}$ from a factored form?
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Find holes, VAs, HAs/slant, intercepts, then sketch behavior. Systematic approach ensures all features are identified.
Find holes, VAs, HAs/slant, intercepts, then sketch behavior. Systematic approach ensures all features are identified.
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Find the hole point for $f(x)=\frac{x^2-4}{x-2}$.
Find the hole point for $f(x)=\frac{x^2-4}{x-2}$.
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Hole at $\left(2,4\right)$. Substitute $x=2$ into simplified form $x+2$.
Hole at $\left(2,4\right)$. Substitute $x=2$ into simplified form $x+2$.
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Identify whether $f(x)=\frac{x^2-4}{x-2}$ has a hole or vertical asymptote at $x=2$.
Identify whether $f(x)=\frac{x^2-4}{x-2}$ has a hole or vertical asymptote at $x=2$.
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A hole at $x=2$. Factor $(x-2)$ cancels from numerator $(x^2-4)$.
A hole at $x=2$. Factor $(x-2)$ cancels from numerator $(x^2-4)$.
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What is the slant asymptote of $f(x)=\frac{x^2-4}{x-2}$?
What is the slant asymptote of $f(x)=\frac{x^2-4}{x-2}$?
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$y=x+2$. Factor and cancel: $\frac{x^2-4}{x-2}=x+2$ after division.
$y=x+2$. Factor and cancel: $\frac{x^2-4}{x-2}=x+2$ after division.
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Identify the vertical asymptote of $f(x)=\frac{x^2+1}{x}$.
Identify the vertical asymptote of $f(x)=\frac{x^2+1}{x}$.
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$x=0$. Denominator is zero when $x=0$.
$x=0$. Denominator is zero when $x=0$.
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What is the slant asymptote of $f(x)=\frac{x^2+1}{x}$?
What is the slant asymptote of $f(x)=\frac{x^2+1}{x}$?
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$y=x$. Polynomial division: $\frac{x^2+1}{x}=x+\frac{1}{x}$.
$y=x$. Polynomial division: $\frac{x^2+1}{x}=x+\frac{1}{x}$.
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Identify the simplified function for $f(x)=\frac{x^2-1}{x-1}$ for $x\neq 1$.
Identify the simplified function for $f(x)=\frac{x^2-1}{x-1}$ for $x\neq 1$.
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$f(x)=x+1$ for $x\neq 1$. After canceling $(x-1)$: $\frac{x^2-1}{x-1}=x+1$.
$f(x)=x+1$ for $x\neq 1$. After canceling $(x-1)$: $\frac{x^2-1}{x-1}=x+1$.
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Find the hole point for $f(x)=\frac{x^2-1}{x-1}$.
Find the hole point for $f(x)=\frac{x^2-1}{x-1}$.
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Hole at $\left(1,2\right)$. Substitute $x=1$ into simplified form $x+1$.
Hole at $\left(1,2\right)$. Substitute $x=1$ into simplified form $x+1$.
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Identify whether $f(x)=\frac{x^2-1}{x-1}$ has a hole or vertical asymptote at $x=1$.
Identify whether $f(x)=\frac{x^2-1}{x-1}$ has a hole or vertical asymptote at $x=1$.
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A hole at $x=1$. Factor $(x-1)$ cancels from numerator and denominator.
A hole at $x=1$. Factor $(x-1)$ cancels from numerator and denominator.
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Identify the horizontal asymptote of $f(x)=\frac{x^2-9}{x^2-4x+4}$.
Identify the horizontal asymptote of $f(x)=\frac{x^2-9}{x^2-4x+4}$.
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$y=1$. Equal degrees: $\frac{1}{1}=1$.
$y=1$. Equal degrees: $\frac{1}{1}=1$.
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Identify the horizontal asymptote of $f(x)=\frac{2x^3}{-4x^3+7}$.
Identify the horizontal asymptote of $f(x)=\frac{2x^3}{-4x^3+7}$.
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$y=-\frac{1}{2}$. Leading coefficient ratio: $\frac{2}{-4}=-\frac{1}{2}$.
$y=-\frac{1}{2}$. Leading coefficient ratio: $\frac{2}{-4}=-\frac{1}{2}$.
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What is the horizontal asymptote of $f(x)=\frac{-7x^4+1}{2x^4-3x}$?
What is the horizontal asymptote of $f(x)=\frac{-7x^4+1}{2x^4-3x}$?
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$y=-\frac{7}{2}$. Equal degrees: $\frac{-7}{2}=-\frac{7}{2}$.
$y=-\frac{7}{2}$. Equal degrees: $\frac{-7}{2}=-\frac{7}{2}$.
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What is the end behavior of $f(x)=\frac{4x^3-x}{2x^3+7}$ as $x\to\pm\infty$?
What is the end behavior of $f(x)=\frac{4x^3-x}{2x^3+7}$ as $x\to\pm\infty$?
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$f(x)\to 2$. Equal degrees: $\frac{4}{2}=2$.
$f(x)\to 2$. Equal degrees: $\frac{4}{2}=2$.
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What is the end behavior of $f(x)=\frac{3x^2+1}{x^3-5}$ as $x\to\pm\infty$?
What is the end behavior of $f(x)=\frac{3x^2+1}{x^3-5}$ as $x\to\pm\infty$?
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$f(x)\to 0$. Numerator degree less than denominator degree.
$f(x)\to 0$. Numerator degree less than denominator degree.
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Identify the vertical asymptote of $f(x)=\frac{x+2}{x-1}$.
Identify the vertical asymptote of $f(x)=\frac{x+2}{x-1}$.
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$x=1$. Denominator is zero when $x-1=0$.
$x=1$. Denominator is zero when $x-1=0$.
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