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Understanding Parts of a Hyperbola in Standard Equations

Learn about the various parts of a hyperbola, including center, foci, vertices, asymptotes, and equations, with examples and graphing explanations.

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Understanding Parts of a Hyperbola in Standard Equations

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  1. HYPERBOLAS Difference Foci Center Vertex Asymptotes Transverse Axis Conjugate Axis

  2. y b c transverse axis x a conjugate axis F F 1 2 Standard 4, 9,16, 17 PARTS OF A HYPERBOLA asymptote asymptote vertex vertex

  3. y - 2 2 2 c = a + b x y 2 2 (x – h) (y – k) = 1 2 2 (y – k) (x – h) = 1 x - 2 2 a b 2 2 a b For both equations. F F F F 1 2 2 1 Standard 4, 9, 16, 17 STANDARD EQUATIONS OF A HYPERBOLA • Hyperbola with center at (h,k) with horizontal axis • has equation In this case, transverse axis is horizontal. • Hyperbola with center at (h,k) with vertical axis has equation In this case, transverse axis is vertical. NOTE: These two hyperbolas are graphed with center (0,0)

  4. y 10 2 2 2 8 c = a + b 6 4 2 -10 x -8 -6 -4 -2 2 8 6 10 4 -2 -4 -6 -8 2 c = 25 + 9 5.8 c= 34 2 2 (x -(+3)) (y-(+4)) - =1 9 25 2 2 c = 34 b = 9 2 a = 25 2 2 (x-3) (y-4) - =1 9 25 Standard 4, 9, 16, 17 Draw the hyperbola that is represented by: h= 3 Center = (3,4) k= 4 a = 5 b= 3 Focus 1= (h+c, k) = (3+5.8,4) = (8.8,4) = (3-5.8,4) Focus 2= (h-c, k) = (-2.8,4)

  5. y 20 16 2 2 2 c = a + b 12 8 4 -20 x -16 -12 -8 -4 16 12 20 8 4 -4 -8 -12 -16 2 c = 81 + 64 12 c= 145 2 2 (y-(+6)) (x-(-2)) - =1 64 81 2 2 c = 145 b = 64 2 a = 81 2 2 (y-6) (x+2) - =1 64 81 Standard 4, 9, 16, 17 Draw the hyperbola that is represented by: h= -2 Center = (-2,6) k= 6 a = 9 b= 8 Focus 1= (h+c, k) = (-2,6+12) = (-2,18) = (-2,6-12) Focus 2= (h-c, k) = (-2,-6)

  6. y 10 2 2 2 8 c = a + b 13 13 13 13 13 13 2 2 6 -a -a 4 2 2 2 b = c - a 2 (-1,0) -10 x -8 -6 -4 -2 2 8 6 10 4 (2,0) -2 -4 -6 -8 (5,0) 2 2 2 2 -9 b = (x – h) (y – k) (2- , 0) (2+ , 0) (2+ , 0) = 1 If = 2 b = 13-9 2 2 (x-(+2)) (y-(0)) 2 2 2+c 2+c - - =1 2 a 4 9 2 b = 4 2 b 2 a = 9 2 2 (x-2) y - =1 2 + c = 2 + c = 4 9 Standard 4, 9, 16, 17 Given the graph below obtain the equation of the hyperbola. Transverse axis is 6 units: 2a=6 a = 3 Focus2 Focus1 then we know: From the figure: h= 2 Center = (2,0) k= 0 Hyperbola is horizontal: Focus1 =( h+c,k) then -2 -2

  7. y 10 2 2 2 8 c = a + b 13 13 13 13 13 13 2 2 6 -a -a 4 2 2 2 b = c - a 2 -10 x -8 -6 -4 -2 2 8 6 10 4 -2 -4 (4,-2) -6 -8 2 2 2 2 -4 b = (y – k) (x – h) = 1 If 2 b = 13-4 (4,-2- ) (4,-2+ ) 2 2 (y-(-2)) (y-(4)) 2 2 -2+c -2+c - - =1 2 a 9 4 2 b = 9 = (4,-2+ ) 2 b 2 a = 4 2 2 (y+2) (x-4) - =1 -2 + c = -2 + c = 9 4 Standard 4, 9,16, 17 Given the graph below obtain the equation of the hyperbola. Transverse axis is 4 units: Focus1 2a=4 a = 2 then we know: Focus2 From the figure: h= 4 Center = (4,-2) k= -2 Focus1 =( h,k+c) Hyperbola is vertical: then +2 +2

  8. 2 Ax + Bxy + Cy + Dx + Ex + F = 0 2 Standard 4, 9, 16, 17 Equation of a Conic Section

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