1 LC The Hyperbola (Algebraic Perspective) MCR3U - Santowski

2 (A) Review The standard equation for a hyperbola is x 2 /a 2 - y 2 /b 2 = 1 (where the 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) The standard equation for a hyperbola is x 2 /a 2 - y 2 /b 2 = 1 (where the 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) 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, and the equation becomes x 2 /b 2 - y 2 /a 2 = -1 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, and the equation becomes x 2 /b 2 - y 2 /a 2 = -1 In a hyperbola, the “minor axis” is referred to as the conjugate axis, but is not really a part of the graph of the hyperbola In a hyperbola, the “minor axis” is referred to as the conjugate axis, but is not really a part of the graph of the hyperbola The intercepts of our hyperbola are at +a (opening L/R) The intercepts of our hyperbola are at +a (opening L/R) The vertices of the hyperbola are at +a and the length of the transverse (major) axis is 2a The vertices of the hyperbola are at +a and the length of the transverse (major) axis is 2a The domain is -a>x>a and range is yER for hyperbola opening L/R The domain is -a>x>a and range is yER for hyperbola opening L/R The two foci are located at (+c,0) for opening L/R The two foci are located at (+c,0) for opening L/R The asymptotes of the hyperbola are at y = (+b/a)x for opening L/R The asymptotes of the hyperbola are at y = (+b/a)x for opening L/R NEW POINT  the foci are related to the values of a and b by the relationship that c 2 = a 2 + b 2 NEW POINT  the foci are related to the values of a and b by the relationship that c 2 = a 2 + b 2

3 (B) Translating Hyperbolas So far, we have considered hyperbolas from a geometric perspective |PF 1 - PF 2 | = 2a and we have centered the hyperbolas at (0,0) So far, we have considered hyperbolas from a geometric perspective |PF 1 - PF 2 | = 2a and we have centered the hyperbolas at (0,0) Now, if the hyperbola were translated left, right, up, or down, then we make the following adjustment on the equation: Now, if the hyperbola were translated left, right, up, or down, then we make the following adjustment on the equation: So now our centrally located hyperbola has been moved to a new center at (h,k) So now our centrally located hyperbola has been moved to a new center at (h,k)

4 (C) Translating Hyperbolas – An Example Given the hyperbola determine the center, the vertices, the foci, the intercepts and the asymptotes. Then graph Given the hyperbola determine the center, the vertices, the foci, the intercepts and the asymptotes. Then graph The center is clearly at (3,-4)  so our hyperbola was translated from being centered at (0,0) by moving right 3 and down 4  so all major points and features must also have been translated R3 and D4 The center is clearly at (3,-4)  so our hyperbola was translated from being centered at (0,0) by moving right 3 and down 4  so all major points and features must also have been translated R3 and D4 The transverse axis is on the x-axis so the hyperbola opens L/R The transverse axis is on the x-axis so the hyperbola opens L/R The value of a = 4 and b = 5 The value of a = 4 and b = 5 So the original vertices were (+4,0)  the new vertices are (-1,- 4) and (7,-4) So the original vertices were (+4,0)  the new vertices are (-1,- 4) and (7,-4) The endpoints of the “minor” axis were (0,+5)  these have now moved to (3,1), (3,-9) The endpoints of the “minor” axis were (0,+5)  these have now moved to (3,1), (3,-9) The original foci were at ( ) = +6.4  so at (+6.4,0) which have now moved to (-3.4,-4) and (9.4,-4) The original foci were at ( ) = +6.4  so at (+6.4,0) which have now moved to (-3.4,-4) and (9.4,-4) The asymptotes used to be the lines y = +1.25x, which have now moved to y = +1.25(x – 3) - 4 The asymptotes used to be the lines y = +1.25x, which have now moved to y = +1.25(x – 3) - 4

5 (C) Translating Hyperbolas – The Intercepts So no y-intercepts

6 (C) Translating Hyperbolas – The Graph

7 (D) In-Class Examples Ex 1. Graph and find the equation of the hyperbola whose 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 transverse axis is parallel to the y axis (so it opens U/D) Ex 1. Graph and find the equation of the hyperbola whose 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 transverse axis is parallel to the y axis (so it opens U/D) So 2a = 16, so a = 8 So 2a = 16, so a = 8 And 2b = 10, thus b = 5 And 2b = 10, thus b = 5 The asymptotes were at y = (+8/5)x (Since the hyperbola opens U/D, the asymptotes are at y = (+a/b)x) The asymptotes were at y = (+8/5)x (Since the hyperbola opens U/D, the asymptotes are at y = (+a/b)x) And c 2 = a 2 + b 2 = = 89  c = +9.4 And c 2 = a 2 + b 2 = = 89  c = +9.4 Therefore our non-translated points are (0,+8), (+5,0) and (0,+9.4)  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) Therefore our non-translated points are (0,+8), (+5,0) and (0,+9.4)  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) Our equation becomes (x-2) 2 /25 - (y+3) 2 /64 = -1 Our equation becomes (x-2) 2 /25 - (y+3) 2 /64 = -1

8 (D) In-Class Examples – The Graph

9 (E) Internet Links perbola/EquationHyperbola.html - an interactive applet fom AnalyzeMath perbola/EquationHyperbola.html - an interactive applet fom AnalyzeMath perbola/EquationHyperbola.html perbola/EquationHyperbola.html - Examples and explanations from OJK's Precalculus Study Page - Examples and explanations from OJK's Precalculus Study Page s/1314/Hyperbolas.asp - Ellipses from Paul Dawkins at Lamar University s/1314/Hyperbolas.asp - Ellipses from Paul Dawkins at Lamar University s/1314/Hyperbolas.asp s/1314/Hyperbolas.asp l - Graphs of ellipses from WebMath.com l - Graphs of ellipses from WebMath.com l l

10 (F) Homework AW, p540, Q3abc, 5cd, 8, 17, 23 AW, p540, Q3abc, 5cd, 8, 17, 23 Nelson text, p616, Q1- 5eol,7,12,15,16 Nelson text, p616, Q1- 5eol,7,12,15,16