22.5E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{13}\) Graph a Hyperbola with Center at \((0,0)\)
In the following exercises, graph.
- \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\)
- \(\frac{x^{2}}{25}-\frac{y^{2}}{9}=1\)
- \(\frac{x^{2}}{16}-\frac{y^{2}}{25}=1\)
- \(\frac{x^{2}}{9}-\frac{y^{2}}{36}=1\)
- \(\frac{y^{2}}{25}-\frac{x^{2}}{4}=1\)
- \(\frac{y^{2}}{36}-\frac{x^{2}}{16}=1\)
- \(16 y^{2}-9 x^{2}=144\)
- \(25 y^{2}-9 x^{2}=225\)
- \(4 y^{2}-9 x^{2}=36\)
- \(16 y^{2}-25 x^{2}=400\)
- \(4 x^{2}-16 y^{2}=64\)
- \(9 x^{2}-4 y^{2}=36\)
- Answer
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Exercise \(\PageIndex{14}\) Graph a Hyperbola with Center at \((h,k)\)
In the following exercises, graph.
- \(\frac{(x-1)^{2}}{16}-\frac{(y-3)^{2}}{4}=1\)
- \(\frac{(x-2)^{2}}{4}-\frac{(y-3)^{2}}{16}=1\)
- \(\frac{(y-4)^{2}}{9}-\frac{(x-2)^{2}}{25}=1\)
- \(\frac{(y-1)^{2}}{25}-\frac{(x-4)^{2}}{16}=1\)
- \(\frac{(y+4)^{2}}{25}-\frac{(x+1)^{2}}{36}=1\)
- \(\frac{(y+1)^{2}}{16}-\frac{(x+1)^{2}}{4}=1\)
- \(\frac{(y-4)^{2}}{16}-\frac{(x+1)^{2}}{25}=1\)
- \(\frac{(y+3)^{2}}{16}-\frac{(x-3)^{2}}{36}=1\)
- \(\frac{(x-3)^{2}}{25}-\frac{(y+2)^{2}}{9}=1\)
- \(\frac{(x+2)^{2}}{4}-\frac{(y-1)^{2}}{9}=1\)
- Answer
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Exercise \(\PageIndex{15}\) Graph a Hyperbola with Center at \((h,k)\)
In the following exercises,
- Write the equation in standard form and
- Graph.
- \(9 x^{2}-4 y^{2}-18 x+8 y-31=0\)
- \(16 x^{2}-4 y^{2}+64 x-24 y-36=0\)
- \(y^{2}-x^{2}-4 y+2 x-6=0\)
- \(4 y^{2}-16 x^{2}-24 y+96 x-172=0\)
- \(9 y^{2}-x^{2}+18 y-4 x-4=0\)
- Answer
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- \(\frac{(x-1)^{2}}{4}-\frac{(y-1)^{2}}{9}=1\)
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- \(\frac{(y-2)^{2}}{9}-\frac{(x-1)^{2}}{9}=1\)
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- \(\frac{(y+1)^{2}}{1}-\frac{(x+2)^{2}}{9}=1\)
Exercise \(\PageIndex{16}\) Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.
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- \(x=-y^{2}-2 y+3\)
- \(9 y^{2}-x^{2}+18 y-4 x-4=0\)
- \(9 x^{2}+25 y^{2}=225\)
- \(x^{2}+y^{2}-4 x+10 y-7=0\)
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- \(x=-2 y^{2}-12 y-16\)
- \(x^{2}+y^{2}=9\)
- \(16 x^{2}-4 y^{2}+64 x-24 y-36=0\)
- \(16 x^{2}+36 y^{2}=576\)
- Answer
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- Parabola
- Circle
- Hyperbola
- Ellipse
Exercise \(\PageIndex{17}\) Mixed Practice
In the following exercises, graph each equation.
- \(\frac{(y-3)^{2}}{9}-\frac{(x+2)^{2}}{16}=1\)
- \(x^{2}+y^{2}-4 x+10 y-7=0\)
- \(y=(x-1)^{2}+2\)
- \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
- \((x+2)^{2}+(y-5)^{2}=4\)
- \(9 x^{2}-4 y^{2}+54 x+8 y+41=0\)
- \(x=-y^{2}-2 y+3\)
- \(16 x^{2}+9 y^{2}=144\)
- Answer
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Exercise \(\PageIndex{18}\) Writing Exercises
- In your own words, define a hyperbola and write the equation of a hyperbola centered at the origin in standard form. Draw a sketch of the hyperbola labeling the center, vertices, and asymptotes.
- Explain in your own words how to create and use the rectangle that helps graph a hyperbola.
- Compare and contrast the graphs of the equations \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{4}=1\).
- Explain in your own words, how to distinguish the equation of an ellipse with the equation of a hyperbola.
- Answer
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2. Answers may vary
4. Answers may vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?