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You are given the equation of a right hyperbola  \\[\\simplify[all]{y={k}/(x-{a})+{b}}.\\]

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Identifying some of the basic properties (intercepts, asymptotes, quadrants) of a right hyperbola.

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The vertical asymptote corresponds to the value of \$x\$ that results in attempting to divide by \$0\$. For the equation

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\\[\\simplify[all]{y={k}/(x-{a})+{b}}\\]

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This means the equation of the vertical asymptote is \$x=\\var{a}\$.

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The horizontal asymptote corresponds to the value of \$y\$ that results from \$x\$ approaching infinity. As \$x\$ gets really really large, the fraction \$\\simplify{{k}/(x-{a})}\$ gets really really close to zero. The bigger \$x\$ gets, the closer \$\\simplify{{k}/(x-{a})}\$ gets to zero, and the closer \$\\simplify[all]{y={k}/(x-{a})+{b}}\$ gets to \$y=\\var{b}\$.

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This means the equation of the horizontal asymptote is \$y=\\var{b}\$.

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Notice that in our equation, \$\\var{k}\$ is multiplying the fraction \$\\simplify[all]{1/(x-{a})}\$. It is a general fact that because \$\\var{k}\$ is positive the graph will be in the top right and bottom left parts of the plane.  negative the graph will be in the top left and bottom right parts of the plane. We can see this as follows:

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• By substituting into the equation an \$x\$ value that is to the right of the vertical asymptote \$x=\\var{a}\$, we will find the resulting \$y\$ value is above below the horizontal asymptote \$y=\\var{b}\$.
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• By substituting into the equation an \$x\$ value that is to the left of the vertical asymptote \$x=\\var{a}\$, we will find the resulting \$y\$ value is above below the horizontal asymptote \$y=\\var{b}\$.
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To find the \$y\$-intercept, let \$x=0\$ and solve for \$y\$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$y\$ \$=\$ \$\\simplify[!collectNumbers]{{k}/(0-{a})+{b}}\$ \$=\$ \$\\simplify[fractionNumbers]{{-k/a+b}}\$
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To find the \$x\$-intercept, let \$y=0\$ and solve for \$x\$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$0\$ \$=\$ \$\\simplify{{k}/(x-{a})+{b}}\$ \$\\var{-b}\$ \$=\$ \$\\simplify{{k}/(x-{a})}\$ \$\\simplify{{-b}(x-{a})}\$ \$=\$ \$\\var{k}\$ \$\\simplify{x-{a}}\$ \$=\$ \$\\simplify{{k}/({-b})}\$ \$x\$ \$=\$ \$\\simplify[fractionNumbers]{{-k/b+a}}\$
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The graph of this equation has a vertical asymptote at \$x=\$ [[0]].

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The graph of this equation has a horizontal asymptote at \$y=\$ [[0]].

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The two asymptotes break the plane up into four parts: top right, top left, bottom left and bottom right.

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The graph is in which of these parts of the plane?

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top left

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top right

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bottom left

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bottom right

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The \$y\$-intercept is at \$y=\$[[0]].

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The \$x\$-intercept is at \$x=\$ [[0]].

\n

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