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Five-Minute Check (over Lesson 7-2)
Then/Now
New Vocabulary
Key Concept:Standard Forms of Equations for Hyperbolas
Example 1:Graph Hyperbolas in Standard Form
Example 2:Graph a Hyperbola
Example 3:Write Equations Given Characteristics
Example 4:Find the Eccentricity of a Hyperbola
Key Concept:Classify Conics Using the Discriminant
Example 5:Identify Conic Sections
Example 6: Real-World Example: Apply Hyperbolas
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Over Lesson 7-2
Graph the ellipse given by 4x 2 + y
2 + 16x – 6y – 39 = 0.
A. B.
C. D.
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Over Lesson 7-2
Write an equation in standard form for the ellipse with vertices (–3, –1) and (7, –1) and foci (–2, –1) and (6, –1).
A.
B.
C.
D.
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Over Lesson 7-2
A. 0.632
B. 0.775
C. 0.845
D. 1.290
Determine the eccentricity of the ellipse given by
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Over Lesson 7-2
Write an equation in standard form for a circle with center at (–2, 5) and radius 3.
A. (x + 2)2 + (y – 5)2 = 3
B. (x + 2)2 + (y – 5)2 = 9
C. (x – 2)2 + (y + 5)2 = 9
D. (x – 2)2 + (y + 5)2 = 3
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Over Lesson 7-2
Identify the conic section represented by 8x
2 + 5y 2 – x + 6y = 0.
A. circle
B. ellipse
C. parabola
D. none of the above
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You analyzed and graphed ellipses and circles. (Lesson 7-2)
• Analyze and graph equations of hyperbolas.
• Use equations to identify types of conic sections.
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• hyperbola
• transverse axis
• conjugate axis
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Graph Hyperbolas in Standard Form
A. Graph the hyperbola given by
The equation is in standard form with h = 0, k = 0,
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Graph Hyperbolas in Standard Form
Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola.
Answer:
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Graph Hyperbolas in Standard Form
The equation is in standard form with h = 2 and k = –4. Because a2 = 4 and b2 = 9, a = 2 and b = 3. Use the values of a and b to find c.
c2 = a2 + b2 Equation relating a, b, and cfor a hyperbola
c2 = 4 + 9 a2 = 4 and b2 = 9
B. Graph the hyperbola given by
Solve for c.
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Graph Hyperbolas in Standard Form
Use h, k, a, b, and c to determine the characteristics of the hyperbola.
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Graph Hyperbolas in Standard Form
Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola.
Answer:
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Graph the hyperbola given by
A. B.
C. D.
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Graph a Hyperbola
Graph the hyperbola given by 4x2 – y2 + 24x + 4y = 28.
First, write the equation in standard form.
4x2 – y2 + 24x + 4y = 28 Original equation
4x2 + 24x – y2 + 4y = 28 Isolate and grouplike terms.
4(x2 + 6x) – (y2 – 4y) = 28 Factor.
4(x2 + 6x + 9) – (y2 – 4y + 4) = 28 + 4(9) – 4Complete thesquares.
4(x + 3)2 – (y – 2)2 = 60 Factor andsimplify.
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Divide each side by 60.
Graph a Hyperbola
The equation is now in standard form with h = –3,
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Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola.
Graph a Hyperbola
Answer:
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Graph the hyperbola given by 3x2 – y2 – 30x – 4y = –119.
A. B.
C. D.
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Write Equations Given Characteristics
A. Write an equation for the hyperbola with foci (1, –5) and (1, 1) and transverse axis length of 4 units.
Because the x-coordinates of the foci are the same, the transverse axis is vertical. Find the center and the values of a, b, and c.
center: (1, –2) Midpoint of segmentbetween foci
a = 2 Transverse axis = 2a
c = 3 Distance from each focus to center
c2 = a2 + b2
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Write Equations Given Characteristics
Answer:
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Write Equations Given Characteristics
B. Write an equation for the hyperbola with vertices (–3, 10) and (–3, –2) and conjugate axis length of 6 units.
Because the x-coordinates of the foci are the same, the transverse axis is vertical. Find the center and the values of a, b, and c.
center: (–3, 4) Midpoint of segmentbetween vertices
b = 3 Conjugate axis = 2b
a = 6 Distance from each vertexto center
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Write Equations Given Characteristics
Answer:
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Write an equation for the hyperbola with foci at (13, –3) and (–5, –3) and conjugate axis length of 12 units.
A.
B.
C.
D.
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Find the Eccentricity of a Hyperbola
Find c and then determine the eccentricity.
c2 = a2 + b2 Equation relating a, b, and c
c2 = 32 + 25 a2 = 32 and b2 = 25
Simplify.
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Find the Eccentricity of a Hyperbola
The eccentricity of the hyperbola is about 1.33.
Simplify.
Eccentricity equation
Answer: 1.33
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A. 0.59
B. 0.93
C. 1.24
D. 1.69
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Identify Conic Sections
A. Use the discriminant to identify the conic section in the equation 2x2 + y2 – 2x + 5xy + 12 = 0.
A is 2, B is 5, and C is 1.
Find the discriminant.
B2 – 4AC = 52 – 4(2)(1) or 17
The discriminant is greater than 0, so the conic is a hyperbola.
Answer: hyperbola
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Identify Conic Sections
B. Use the discriminant to identify the conic section in the equation 4x2 + 4y2 – 4x + 8 = 0.
A is 4, B is 0, and C is 4.
Find the discriminant.
B2 – 4AC = 02 – 4(4)(4) or –64
The discriminant is less than 0, so the conic must be either a circle or an ellipse. Because A = C, the conic is a circle.
Answer: circle
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Identify Conic Sections
C. Use the discriminant to identify the conic section in the equation 2x2 + 2y2 – 6y + 4xy – 10 = 0.
A is 2, B is 4, and C is 2.
Find the discriminant.
B2 – 4AC = 42 – 4(2)(2) or 0
The discriminant is 0, so the conic is a parabola.
Answer: parabola
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Use the discriminant to identify the conic section given by 15 + 6y + y2 = –14x – 3x2.
A. ellipse
B. circle
C. hyperbola
D. parabola
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Apply Hyperbolas
A. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the equation for the hyperbola on which the ship is located.
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Apply Hyperbolas
First, place the two sensors on a coordinate grid so that the origin is the midpoint of the segment between station E and station F. The ship is closer to station E, so it should be in the 2nd quadrant.
The two stations are located at the foci of the hyperbola, so c is 175. The absolute value of the difference of the distances from any point on a hyperbola to the foci is 2a. Because the ship is 80 miles farther from station F than station E, 2a = 80 and a = 40.
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Apply Hyperbolas
Use the values of a and c to find b2.
c2 = a2 + b2 Equation relating a, b, and c
1752 = 402 + b2 c = 175 and a = 40
29,025 = b2 Simplify.
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Apply Hyperbolas
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Apply Hyperbolas
Answer:
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Apply Hyperbolas
B. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the exact coordinates of the ship if it is 125 miles from the shore.
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Apply Hyperbolas
Original equation
y = 125
Because the ship is 125 miles from the shore, y = 125. Substitute the value of y into the equation and solve for x.
Solve.
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Apply Hyperbolas
Since the ship is closer to station E, it is located on the left branch of the hyperbola, and the value of x is about –49.6. Therefore, the coordinates of the ship are (–49.6, 125).
Answer: (–49.6, 125)
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NAVIGATION Suppose LORAN stations S and T are located 240 miles apart along a straight shore with S due north of T. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 60 miles farther from station T than it is from station S. Find the equation for the hyperbola on which the ship is located.
A.
C.
B.
D.
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