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LC.02.2 - The Hyperbola LC.02.2 - The Hyperbola (Algebraic Perspective)(Algebraic Perspective)

MCR3U - SantowskiMCR3U - Santowski

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(A) Review(A) Review The standard equation for a hyperbola is xThe standard equation for a hyperbola is x22/a/a22 - y - y22/b/b22 = 1 (where the = 1 (where the

hyperbola opens left/right/along the x-axis and the foci on the x-axis hyperbola opens left/right/along the x-axis and the foci on the x-axis and where the transverse (major) axis is on the x-axis)and where the transverse (major) axis is on the x-axis)

Alternatively, if the foci are on the y-axis, and the transverse (major) Alternatively, if the foci are on the y-axis, and the transverse (major) axis is on the y-axis, the hyperbola opens up/down/along the y-axis, axis is on the y-axis, the hyperbola opens up/down/along the y-axis, and the equation becomes xand the equation becomes x22/b/b22 - y - y22/a/a22 = -1 = -1

In a hyperbola, the “minor axis” is referred to as the conjugate axis, In a hyperbola, the “minor axis” is referred to as the conjugate axis, but is not really a part of the graph of the hyperbolabut is not really a part of the graph of the hyperbola

The intercepts of our hyperbola are at The intercepts of our hyperbola are at ++a (opening L/R)a (opening L/R) The vertices of the hyperbola are at The vertices of the hyperbola are at ++a and the length of the a and the length of the

transverse (major) axis is 2atransverse (major) axis is 2a The domain is -a>x>a and range is yER for hyperbola opening L/RThe domain is -a>x>a and range is yER for hyperbola opening L/R The two foci are located at (The two foci are located at (++c,0) for opening L/Rc,0) for opening L/R The asymptotes of the hyperbola are at y = (The asymptotes of the hyperbola are at y = (++b/a)x for opening L/Rb/a)x for opening L/R NEW POINT NEW POINT the foci are related to the values of a and b by the the foci are related to the values of a and b by the

relationship that crelationship that c22 = a = a22 + b + b22

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(B) Translating Hyperbolas(B) Translating Hyperbolas So far, we have considered hyperbolas from a So far, we have considered hyperbolas from a

geometric perspective |PFgeometric perspective |PF11 - PF - PF22| = 2a and we | = 2a and we have centered the hyperbolas at (0,0)have centered the hyperbolas at (0,0)

Now, if the hyperbola were translated left, right, Now, if the hyperbola were translated left, right, up, or down, then we make the following up, or down, then we make the following adjustment on the equation:adjustment on the equation:

So now our centrally located hyperbola has been So now our centrally located hyperbola has been moved to a new center at (h,k)moved to a new center at (h,k)

x h

a

y h

b

2

2

2

2 1

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(C) Translating Hyperbolas – An Example(C) Translating Hyperbolas – An Example

Given the hyperbola determine the Given the hyperbola determine the center, the vertices, the foci, the intercepts and the asymptotes. center, the vertices, the foci, the intercepts and the asymptotes. Then graph Then graph

The center is clearly at (3,-4) The center is clearly at (3,-4) so our hyperbola was translated so our hyperbola was translated from being centered at (0,0) by moving right 3 and down 4 from being centered at (0,0) by moving right 3 and down 4 so so all major points and features must also have been translated R3 all major points and features must also have been translated R3 and D4and D4

The transverse axis is on the x-axis so the hyperbola opens L/RThe transverse axis is on the x-axis so the hyperbola opens L/R The value of a = 4 and b = 5The value of a = 4 and b = 5 So the original vertices were (So the original vertices were (++4,0) 4,0) the new vertices are (-1,-4) the new vertices are (-1,-4)

and (7,-4) and (7,-4) The endpoints of the “minor” axis were (0,The endpoints of the “minor” axis were (0,++5) 5) these have now these have now

moved to (3,1), (3,-9) moved to (3,1), (3,-9) The original foci were at The original foci were at (5(522 + 4 + 42)2) = = ++6.4 6.4 so at ( so at (++6.4,0) which 6.4,0) which

have now moved to (-3.4,-4) and (9.4,-4)have now moved to (-3.4,-4) and (9.4,-4) The asymptotes used to be the lines y = The asymptotes used to be the lines y = ++1.25x, which have now 1.25x, which have now

moved to y = moved to y = ++1.25(x – 3) - 41.25(x – 3) - 4

x y

3

16

4

251

2 2

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(C) Translating Hyperbolas – The Intercepts(C) Translating Hyperbolas – The Intercepts

So no y-interceptsSo no y-intercepts

( ) ( )

( ) ( )

( ) .

.

0 3

16

4

251

25 9 16 4 400

4400 225

1610 9375

4 10 9375

2 2

2

2

y

y

y

y

y R

( ) ( )

( ) ( )

( ) .

.

.

.

x

x

x

x

x

3

16

0 4

251

25 3 16 16 400

3400 256

2526 24

3 5 12

2 12

8 12

2 2

2

2

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(C) Translating Hyperbolas – The Graph(C) Translating Hyperbolas – The Graph

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(D) In-Class Examples(D) In-Class Examples Ex 1. Graph and find the equation of the hyperbola whose Ex 1. Graph and find the equation of the hyperbola whose

transverse axis has a length of 16 and whose conjugate transverse axis has a length of 16 and whose conjugate axis has a length of 10 units. Its center is at (2,-3) and the axis has a length of 10 units. Its center is at (2,-3) and the transverse axis is parallel to the y axis (so it opens U/D)transverse axis is parallel to the y axis (so it opens U/D)

So 2a = 16, so a = 8So 2a = 16, so a = 8 And 2b = 10, thus b = 5And 2b = 10, thus b = 5 The asymptotes were at y = (The asymptotes were at y = (++8/5)x (Since the hyperbola 8/5)x (Since the hyperbola

opens U/D, the asymptotes are at y = (opens U/D, the asymptotes are at y = (++a/b)x)a/b)x) And cAnd c22 = a = a22 + b + b22 = 64 + 25 = 89 = 64 + 25 = 89 c = c = ++9.49.4 Therefore our non-translated points are (0,Therefore our non-translated points are (0,++8), (8), (++5,0) and 5,0) and

(0,(0,++9.4) 9.4) now translating them by R2 and D3 gives us now translating them by R2 and D3 gives us new points at (2,5),(2-11),(-3,-3),(7,-3),(2,-12.4),(2,6.4)new points at (2,5),(2-11),(-3,-3),(7,-3),(2,-12.4),(2,6.4)

Our equation becomes (x-2)Our equation becomes (x-2)22/25 - (y+3)/25 - (y+3)22/64 = -1/64 = -1

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(D) In-Class Examples – The (D) In-Class Examples – The GraphGraph

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(E) Internet Links(E) Internet Links

http://www.analyzemath.com/EquationHyphttp://www.analyzemath.com/EquationHyperbola/EquationHyperbola.htmlerbola/EquationHyperbola.html - an interactive applet fom AnalyzeMath - an interactive applet fom AnalyzeMath

http://home.alltel.net/okrebs/page63.htmlhttp://home.alltel.net/okrebs/page63.html - Examples and explanations from OJK's - Examples and explanations from OJK's Precalculus Study PagePrecalculus Study Page

http://tutorial.math.lamar.edu/AllBrowsers/http://tutorial.math.lamar.edu/AllBrowsers/1314/Hyperbolas.asp1314/Hyperbolas.asp - Ellipses from Paul Dawkins at Lamar - Ellipses from Paul Dawkins at Lamar UniversityUniversity

http://www.webmath.com/hyperbolas.htmlhttp://www.webmath.com/hyperbolas.html - Graphs of ellipses from WebMath.com- Graphs of ellipses from WebMath.com

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(F) Homework(F) Homework

AW, p540, Q3abc, 5cd, 8, 17, 23AW, p540, Q3abc, 5cd, 8, 17, 23

Nelson text, p616, Q1-Nelson text, p616, Q1-5eol,7,12,15,165eol,7,12,15,16