22.5E: Exercises

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Aug 13, 2020
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- 22.5: Hyperbolas
- 22.6: Solving Systems of Nonlinear Equations

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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

The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus two-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right. — Figure 11.4.33

The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-fourths times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right. — Figure 11.4.34

The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down. — Figure 11.4.35

The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-fourths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down. — Figure 11.4.36

The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-halves times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down. — Figure 11.4.37

11.

The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right. — Figure 11.4.38

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

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right. — Figure 11.4.39

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, negative 4) an asymptote that passes through (negative 7, 1) and (5, negative 9) and an asymptote that passes through (5, 1) and (negative 7, negative 9), and branches that pass through the vertices (1, 1) and (1, negative 9) and open up and down. — Figure 11.4.41

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 1, 4) an asymptote that passes through (4, 8) and (negative 6, 0) and an asymptote that passes through (negative 6, 8) and (4, 0), and branches that pass through the vertices (negative 1, 0) and (negative 1, 8) and open up and down. — Figure 11.4.42

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (3, negative 2) an asymptote that passes through (8, 1) and (negative 2, negative 5) and an asymptote that passes through (negative 2, negative 1) and (8, negative 5), and branches that pass through the vertices (negative 2, negative 2) and (8, negative 2) and opens left and right. — Figure 11.4.43

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

$\frac{(x-1)^{2}}{4}-\frac{(y-1)^{2}}{9}=1$

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 1) an asymptote that passes through (3, 4) and (negative 1, negative 2) and an asymptote that passes through (negative 1, 4) and (3, negative 2), and branches that pass through the vertices (negative 1, 1) and (3, 1) and opens left and right. — Figure 11.4.44

$\frac{(y-2)^{2}}{9}-\frac{(x-1)^{2}}{9}=1$

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 2) an asymptote that passes through (4, 5) and (negative 2, negative 1) and an asymptote that passes through (negative 2, 5) and (4, negative 1), and branches that pass through the vertices (1, 5) and (1, negative 1) and open up and down. — Figure 11.4.45

$\frac{(y+1)^{2}}{1}-\frac{(x+2)^{2}}{9}=1$

The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 2, negative 1) an asymptote that passes through (1, 0) and (negative 5, negative 2) and an asymptote that passes through (3, 0) and (1, negative 2), and branches that pass through the vertices (negative 2, 0) and (negative 2, negative 2) and open up and down. — Figure 11.4.46

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.

1. $x=-y^{2}-2 y+3$
2. $9 y^{2}-x^{2}+18 y-4 x-4=0$
3. $9 x^{2}+25 y^{2}=225$
4. $x^{2}+y^{2}-4 x+10 y-7=0$
1. $x=-2 y^{2}-12 y-16$
2. $x^{2}+y^{2}=9$
3. $16 x^{2}-4 y^{2}+64 x-24 y-36=0$
4. $16 x^{2}+36 y^{2}=576$

Answer

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

The graph shows the x y coordinate plane with a circle whose center is (2, negative 5) and whose radius is 6 units. — Figure 11.4.47

The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 5) and co-vertices are (plus or minus 3, 0). — Figure 11.4.48

The graph shows the x y coordinate plane with the center (1, 2) an asymptote that passes through (negative 2, 5) and (5, negative 1) and an asymptote that passes through (4, 5) and (2, 0), and branches that pass through the vertices (1, 5) and (negative 2, negative 1) and open up and down. — Figure 11.4.49

The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 4) and co-vertices are (plus or minus 3, 0). — Figure 11.4.50

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

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.

This table has four columns and four rows. The first row is a header and it labels each column, â€œI canâ€¦â€, â€œConfidently,â€ â€œWith some help,â€ and â€œNo-I donâ€™t get it!â€ In row 2, the I can was graph a hyperbola with center at (0, 0). In row 3, the I can was graph a hyperbola with a center at (h, k). In row 4, the I can was identify conic sections by their equations. — Figure 11.4.51

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?

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Practice Makes Perfect

Exercise $\PageIndex{13}$ Graph a Hyperbola with Center at $(0,0)$

Exercise $\PageIndex{14}$ Graph a Hyperbola with Center at $(h,k)$

Exercise $\PageIndex{15}$ Graph a Hyperbola with Center at $(h,k)$

Exercise $\PageIndex{16}$ Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola

Exercise $\PageIndex{17}$ Mixed Practice

Exercise $\PageIndex{18}$ Writing Exercises

Self Check

Practice Makes Perfect

Exercise 22.5E.13\PageIndex{13} Graph a Hyperbola with Center at (0,0)(0,0)

Exercise 22.5E.14\PageIndex{14} Graph a Hyperbola with Center at (h,k)(h,k)

Exercise 22.5E.15\PageIndex{15} Graph a Hyperbola with Center at (h,k)(h,k)

Exercise 22.5E.16\PageIndex{16} Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola

Exercise 22.5E.17\PageIndex{17} Mixed Practice

Exercise 22.5E.18\PageIndex{18} Writing Exercises

Self Check

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Exercise $\PageIndex{13}$ Graph a Hyperbola with Center at $(0,0)$

Exercise $\PageIndex{14}$ Graph a Hyperbola with Center at $(h,k)$

Exercise $\PageIndex{15}$ Graph a Hyperbola with Center at $(h,k)$

Exercise $\PageIndex{16}$ Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola

Exercise $\PageIndex{17}$ Mixed Practice

Exercise $\PageIndex{18}$ Writing Exercises