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$\frac{{\left(y-6\right)}^{2}}{36}-\frac{{\left(x+1\right)}^{2}}{16}=1$
$\frac{{\left(x-2\right)}^{2}}{49}-\frac{{\left(y+7\right)}^{2}}{49}=1$
$\frac{{\left(x-2\right)}^{2}}{{7}^{2}}-\frac{{\left(y+7\right)}^{2}}{{7}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(9,-7\right),\left(-5,-7\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(2+7\sqrt{2},-7\right),\left(2-7\sqrt{2},-7\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=x-9,y=-x-5$
$4{x}^{2}-8x-9{y}^{2}-72y+112=0$
$-9{x}^{2}-54x+9{y}^{2}-54y+81=0$
$\frac{{\left(x+3\right)}^{2}}{{3}^{2}}-\frac{{\left(y-3\right)}^{2}}{{3}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(0,3\right),\left(-6,3\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(-3+3\sqrt{2},1\right),\left(-3-3\sqrt{2},1\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=x+6,y=-x$
$4{x}^{2}-24x-36{y}^{2}-360y+864=0$
$-4{x}^{2}+24x+16{y}^{2}-128y+156=0$
$\frac{{\left(y-4\right)}^{2}}{{2}^{2}}-\frac{{\left(x-3\right)}^{2}}{{4}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(3,6\right),\left(3,2\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(3,4+2\sqrt{5}\right),\left(3,4-2\sqrt{5}\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=\frac{1}{2}\left(x-3\right)+4,y=-\frac{1}{2}\left(x-3\right)+4$
$-4{x}^{2}+40x+25{y}^{2}-100y+100=0$
${x}^{2}+2x-100{y}^{2}-1000y+2401=0$
$\frac{{\left(y+5\right)}^{2}}{{7}^{2}}-\frac{{\left(x+1\right)}^{2}}{{70}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(-1,2\right),\left(-1,-12\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(-1,-5+7\sqrt{101}\right),\left(-1,-5-7\sqrt{101}\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=\frac{1}{10}\left(x+1\right)-5,y=-\frac{1}{10}\left(x+1\right)-5$
$-9{x}^{2}+72x+16{y}^{2}+16y+4=0$
$4{x}^{2}+24x-25{y}^{2}+200y-464=0$
$\frac{{\left(x+3\right)}^{2}}{{5}^{2}}-\frac{{\left(y-4\right)}^{2}}{{2}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(2,4\right),\left(-8,4\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(-3+\sqrt{29},4\right),\left(-3-\sqrt{29},4\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=\frac{2}{5}\left(x+3\right)+4,y=-\frac{2}{5}\left(x+3\right)+4$
For the following exercises, find the equations of the asymptotes for each hyperbola.
$\frac{{y}^{2}}{{3}^{2}}-\frac{{x}^{2}}{{3}^{2}}=1$
$\frac{{\left(x-3\right)}^{2}}{{5}^{2}}-\frac{{\left(y+4\right)}^{2}}{{2}^{2}}=1$
$y=\frac{2}{5}\left(x-3\right)-4,y=-\frac{2}{5}\left(x-3\right)-4$
$\frac{{\left(y-3\right)}^{2}}{{3}^{2}}-\frac{{\left(x+5\right)}^{2}}{{6}^{2}}=1$
$9{x}^{2}-18x-16{y}^{2}+32y-151=0$
$y=\frac{3}{4}\left(x-1\right)+1,y=-\frac{3}{4}\left(x-1\right)+1$
$16{y}^{2}+96y-4{x}^{2}+16x+112=0$
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.
$\frac{{x}^{2}}{64}-\frac{{y}^{2}}{4}=1$
$81{x}^{2}-9{y}^{2}=1$
$\frac{{\left(y+5\right)}^{2}}{9}-\frac{{\left(x-4\right)}^{2}}{25}=1$
$\frac{{\left(x-2\right)}^{2}}{8}-\frac{{\left(y+3\right)}^{2}}{27}=1$
$\frac{{\left(y-3\right)}^{2}}{9}-\frac{{\left(x-3\right)}^{2}}{9}=1$
$-4{x}^{2}-8x+16{y}^{2}-32y-52=0$
$-{x}^{2}+8x+4{y}^{2}-40y+88=0$
$16{x}^{2}+64x-4{y}^{2}-8y-4=0$
$4{x}^{2}+16x-4{y}^{2}+16y+16=0$
For the following exercises, given information about the graph of the hyperbola, find its equation.
Vertices at $\text{\hspace{0.17em}}\left(3,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(\mathrm{-3},0\right)\text{\hspace{0.17em}}$ and one focus at $\text{\hspace{0.17em}}\left(5,0\right).$
$\frac{{x}^{2}}{9}-\frac{{y}^{2}}{16}=1$
Vertices at $\text{\hspace{0.17em}}\left(0,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(0,\mathrm{-6}\right)\text{\hspace{0.17em}}$ and one focus at $\text{\hspace{0.17em}}\left(0,\mathrm{-8}\right).$
Vertices at $\text{\hspace{0.17em}}\left(1,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(11,1\right)\text{\hspace{0.17em}}$ and one focus at $\text{\hspace{0.17em}}\left(12,1\right).$
$\frac{{\left(x-6\right)}^{2}}{25}-\frac{{\left(y-1\right)}^{2}}{11}=1$
Center: $\text{\hspace{0.17em}}\left(0,0\right);$ vertex: $\text{\hspace{0.17em}}\left(0,\mathrm{-13}\right);$ one focus: $\text{\hspace{0.17em}}\left(0,\sqrt{313}\right).$
Center: $\text{\hspace{0.17em}}\left(4,2\right);$ vertex: $\text{\hspace{0.17em}}\left(9,2\right);$ one focus: $\text{\hspace{0.17em}}\left(4+\sqrt{26},2\right).$
$\frac{{\left(x-4\right)}^{2}}{25}-\frac{{\left(y-2\right)}^{2}}{1}=1$
Center: $\text{\hspace{0.17em}}\left(3,5\right);\text{\hspace{0.17em}}$ vertex: $\text{\hspace{0.17em}}\left(3,11\right);\text{\hspace{0.17em}}$ one focus: $\text{\hspace{0.17em}}\left(3,5+2\sqrt{10}\right).$
For the following exercises, given the graph of the hyperbola, find its equation.
$\frac{{\left(x+3\right)}^{2}}{25}-\frac{{\left(y+3\right)}^{2}}{25}=1$
For the following exercises, express the equation for the hyperbola as two functions, with $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.
$\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}=1$
$\frac{{y}^{2}}{9}-\frac{{x}^{2}}{1}=1$
$y\left(x\right)=3\sqrt{{x}^{2}+1},y\left(x\right)=-3\sqrt{{x}^{2}+1}$
$\frac{{\left(x-2\right)}^{2}}{16}-\frac{{\left(y+3\right)}^{2}}{25}=1$
$-4{x}^{2}-16x+{y}^{2}-2y-19=0$
$y\left(x\right)=1+2\sqrt{{x}^{2}+4x+5},y\left(x\right)=1-2\sqrt{{x}^{2}+4x+5}$
$4{x}^{2}-24x-{y}^{2}-4y+16=0$
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.
The hedge will follow the asymptotes $\text{\hspace{0.17em}}y=x\text{and}y=-x,$ and its closest distance to the center fountain is 5 yards.
$\frac{{x}^{2}}{25}-\frac{{y}^{2}}{25}=1$
The hedge will follow the asymptotes $\text{\hspace{0.17em}}y=2x\text{and}y=\mathrm{-2}x,$ and its closest distance to the center fountain is 6 yards.
The hedge will follow the asymptotes $\text{\hspace{0.17em}}y=\frac{1}{2}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=-\frac{1}{2}x,$ and its closest distance to the center fountain is 10 yards.
$\frac{{x}^{2}}{100}-\frac{{y}^{2}}{25}=1$
The hedge will follow the asymptotes $\text{\hspace{0.17em}}y=\frac{2}{3}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=-\frac{2}{3}x,$ and its closest distance to the center fountain is 12 yards.
The hedge will follow the asymptotes $\text{}y=\frac{3}{4}x\text{and}y=-\frac{3}{4}x,$ and its closest distance to the center fountain is 20 yards.
$\frac{{x}^{2}}{400}-\frac{{y}^{2}}{225}=1$
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.
The object enters along a path approximated by the line $\text{\hspace{0.17em}}y=x-2\text{\hspace{0.17em}}$ and passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $\text{\hspace{0.17em}}y=-x+2.\text{\hspace{0.17em}}$
The object enters along a path approximated by the line $\text{\hspace{0.17em}}y=2x-2\text{\hspace{0.17em}}$ and passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $\text{\hspace{0.17em}}y=\mathrm{-2}x+2.\text{\hspace{0.17em}}$
$\frac{{\left(x-1\right)}^{2}}{0.25}-\frac{{y}^{2}}{0.75}=1$
The object enters along a path approximated by the line $\text{\hspace{0.17em}}y=0.5x+2\text{\hspace{0.17em}}$ and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $\text{\hspace{0.17em}}y=\mathrm{-0.5}x-2.\text{\hspace{0.17em}}$
The object enters along a path approximated by the line $\text{\hspace{0.17em}}y=\frac{1}{3}x-1\text{\hspace{0.17em}}$ and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $\text{}y=-\frac{1}{3}x+1.$
$\frac{{\left(x-3\right)}^{2}}{4}-\frac{{y}^{2}}{5}=1$
The object It enters along a path approximated by the line $\text{\hspace{0.17em}}y=3x-9\text{\hspace{0.17em}}$ and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $\text{\hspace{0.17em}}y=\mathrm{-3}x+9.\text{\hspace{0.17em}}$
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