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Hyperbola Graphing and Equation Identification

Learn how to graph a hyperbola, identify its vertices, foci, and asymptotes, and write its equation based on given foci and vertices.

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Hyperbola Graphing and Equation Identification

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  1. 100 y2 25y2 4x2 100 4 100 100 = – y225 – = 1 EXAMPLE 1 Graph an equation of a hyperbola Graph 25y2 – 4x2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola. SOLUTION STEP 1 Rewrite the equation in standard form. 25y2 – 4x2 = 100 Write original equation. Divide each side by 100. Simplify.

  2. 29. 29. soc = The foci are at( 0, + ) (0, + 5.4). 25 ab + + The asymptotes are y = x x y = or EXAMPLE 1 Graph an equation of a hyperbola STEP 2 Identify the vertices, foci, and asymptotes. Note that a2 = 4 and b2 = 25, so a = 2and b = 5. The y2 - term is positive, so the transverse axis is vertical and the vertices are at (0, +2). Find the foci. c2 = a2– b2 = 22– 52 = 29.

  3. EXAMPLE 1 Graph an equation of a hyperbola STEP 3 Draw the hyperbola. First draw a rectangle centered at the origin that is 2a= 4units high and 2b= 10units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.

  4. EXAMPLE 2 Write an equation of a hyperbola Write an equation of the hyperbola with foci at (–4, 0) and (4, 0) and vertices at (–3, 0) and (3, 0). SOLUTION The foci and vertices lie on the x-axis equidistant from the origin, so the transverse axis is horizontal and the center is the origin. The foci are each 4 units from the center, so c = 4. The vertices are each 3 units from the center, soa = 3.

  5. = 1 x2 x2 9 32 = 1 y2 7 y2 7 – – EXAMPLE 2 Write an equation of a hyperbola Becausec2 = a2 + b2,you haveb2 = c2 – a2. Findb2. b2 = c2–a2 = 42–32 = 7 Because the transverse axis is horizontal, the standard form of the equation is as follows: Substitute 3 for aand 7 for b2. Simplify

  6. x2 65 , 0 16 , ( + ) , y2 49 74 + – y = x for Examples 1 and 2 GUIDED PRACTICE Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola. 1. = 1 SOLUTION (+4, 0)

  7. ( 0, + ) , y2 37 36 for Examples 1 and 2 GUIDED PRACTICE 2. = 1 – x2 SOLUTION (0, +6) , y = +6x

  8. ( 0, + ) , 13 32 + y = x for Examples 1 and 2 GUIDED PRACTICE 3. 4y2 – 9x2 = 36 SOLUTION (0, +3) ,

  9. 5.Foci:(0, – 10), (0, 10) Vertices:(0, – 6), (0, 6) 4.Foci:(–3, 0), (3, 0) Vertices:(–1, 0), (1, 0) = 1 y2 y2 8 36 – = 1 x2 x2 64 – for Examples 1 and 2 GUIDED PRACTICE Write an equation of the hyperbola with the given foci and vertices. SOLUTION SOLUTION

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