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C O N I C S E C T I O N S Part 4: Hyperbola. Hyperbola.

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Presentation on theme: "C O N I C S E C T I O N S Part 4: Hyperbola. Hyperbola."— Presentation transcript:

1 C O N I C S E C T I O N S Part 4: Hyperbola

2

3 Hyperbola

4 Hyperbolas (opening left and right) Foci Vertices The transverse axis is the line segment joining the vertices. asymptotes y = (x - h) 2 – (y – k) 2 = 1 a 2 b 2 (–c, 0) (c, 0) (–a, 0) (a, 0) (b, 0) (–b, 0) The foci of the hyperbola lie on the major axis, c units from the center, where c 2 = a 2 + b 2 The major axis is horizontal and acts as the axis of symmetry. It will contain the Vertices and the Foci. The minor axis is vertical and acts as the line of reflection. It will contain the two “b” points. The Vertices are “a” distance from the center. B-points The B-points are “b” distance from the center. The asymptotes can be found using the box method, where the “a” and “b” points help form a box ….. Center: (h,k) Or you can use the linear equations.

5 Hyperbolas (opening up and down) Foci Vertices The transverse axis is the line segment joining the vertices. asymptotes y = (y - k) 2 – (x – h) 2 = 1 a 2 b 2 (–c, 0) (c, 0) (–a, 0) (a, 0) (b, 0) (–b, 0) The foci of the hyperbola lie on the major axis, c units from the center, where c 2 = a 2 + b 2 The major axis is vertical and acts as the axis of symmetry. It will contain the Vertices and the Foci. The minor axis is horizontal and acts as the line of reflection. It will contain the two “b” points. The Vertices are “a” distance from the center. b-points The b-points are “b” distance from the center. The asymptotes can be found using the box method, where the “a” and “b” points help form a box ….. Center: (h,k) Or you can use the linear equations.

6 Example: Write an equation of the hyperbola with foci (0, –6) and (0, 6) and vertices (0, –4) and (0, 4). Its center is (0, 0). (y – h) 2 – (x – k) 2 = 1 a 2 b 2 vertical a = 4, c = 6 c 2 = a 2 + b = b 2 36 = 16 + b 2 20 = b 2 The equation of the hyperbola: y 2 – x 2 = (–b, 0)(b, 0) (0, 4) (0, –4) (0, 6) (0, –6)

7 Example: Graph y 2 – x 2 = 1 ; find foci and asymptotes 9 25 Draw the rectangle and asymptotes... (0,–3) (0, 3) (5, 0)(–5,0) a = 3 b = 5 vertical c 2 = a 2 + b 2 c 2 = = 34 c = Foci: Asymptotes:

8 Example: Write the equation in standard form of 4x 2 – 16y 2 = 64. Find the foci and vertices of the hyperbola. Get the equation in standard form (make it equal to 1): 4x 2 – 16y 2 = Use c 2 = a 2 + b 2 to find c. c 2 = c 2 = = 20 c = (c, 0)(–c,0) (–4,0)(4, 0) (0, 2) (0,-2) That means a = 4 b = 2 Vertices: Foci: Simplify... x 2 – y 2 =

9 vertical Center: (–3, 2) a = 5 b = 4 Example: Graph (y – 2) 2 – (x + 3) 2 = (1, 2)(–7, 2) (–3, 7) (–3, –3) To graph, start with the center… Move 5 units up and down Move 4 units right and left Draw the rectangle and asymptotes…

10 Example: 9x 2 – 4y x + 16y – 43 = 0 9(x + 1) 2 – 4(y – 2) 2 =36 9x x –4y y = 43 9(x 2 + 2x ) – 4(y 2 – 4y ) = – 16 (x + 1) 2 – (y – 2) 2 = a = 2 b = 3 Center (–1, 2) c 2 = a 2 + b 2 = Foci: Asymptotes:

11 Hyperbolas, write equations Example: Write an equation in standard form for the hyperbola with vertices (–1, 1) and (7, 1) and foci (–2, 1) and (8, 1). center: a = b = c = (x – 3) 2 – (y – 1) 2 = (3, 1) 5 c 2 = a 2 + b 2 25 = 16 + b 2 b 2 = 9


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