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Hyperbola Graphing and Equations

Learn how to graph and find the equation of a hyperbola, including its center, vertices, co-vertices, foci, and asymptotes. Explore examples and transformations.

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Hyperbola Graphing and Equations

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  1. 2. 1. x2 y2 x2 y2 25 4 16 9 – = 1 – = 1 Reminder: Multiplying both sides of each equation by the least common multiple eliminates the fractions. 4x2 – 9y2 = 36 16y2 – 25x2 = 400 Objectives Write the standard equation for a hyperbola. Graph a hyperbola, and identify its center, vertices, co-vertices, foci, and asymptotes.

  2. A. B. (y+1)2 (x-3)2 (x-3)2 (y+1)2 x2 y2 16 16 9 4 25 25 + = 1 + = 1 - = 1 Notes 1. Graph the ellipses . 2. Graph the hyperbola . 3. Find the vertices, co-vertices, and asymptotes of , then graph. 4. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0).

  3. What would happen if you pulled the two foci of an ellipse so far apart that they moved outside the ellipse? The result would be a hyperbola, another conic section. A hyperbola is a set of points P(x, y) in a plane such that the difference of the distances from P to fixed points F1 and F2, the foci, is constant. For a hyperbola, d = |PF1 – PF2 |, where d is the constant difference. You can use the distance formula to find the equation of a hyperbola.

  4. The standard form of the equation of a hyperbola depends on whether the hyperbola’s transverse axis is horizontal or vertical.

  5. x2 y2 49 9 – = 1 Example 1 Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph Step 1 The vertices are –7, 0) and (7, 0) and the co-vertices are (0, –3) and (0, 3). Step 2 Draw a box using the vertices and co-vertices. Draw the asymptotes through the corners of the box. Step 3 Draw the hyperbola by using the vertices and the asymptotes.

  6. Step 2 The asymptotes cross at (3, -5) and have a slope = (x –3)2 (y + 5)2 7 9 49 3 – = 1 Example 2: Graphing a Hyperbola Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph. Step 1 Center is (3, -5), the vertices are (0, –5) and (6, –5) and the co-vertices are (3, –12) and (3, 2) .

  7. Example 2B Continued Step 3 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the box. Step 4 Draw the hyperbola by using the vertices and the asymptotes.

  8. x2 y2 x2 y2 a2 b2 36 36 Step 2: Because a = 6 and b = 6, the equation of the graph is Step 1: The graph opens horizontally, so the equation will be in the form of – = 1 – = 1. Example 3A: Writing Equations of Hyperbolas Write an equation in standard form for each hyperbola.

  9. Step 3 The equation of the hyperbola is . x2 y2 Step 1 Because the vertex and the focus are on the horizontal axis 16 84 – = 1 x2 y2 16 b2 – = 1 Example 3B: Writing Equations of Hyperbolas Write an equation for the hyperbola with center at the origin, vertex (4, 0), and focus (10, 0). Step 2 Use focus2 = 16 + other denominator.

  10. A. B. (y+1)2 (x-3)2 (x-3)2 (y+1)2 x2 y2 144 144 9 4 25 25 + = 1 + = 1 - = 1 Notes 1. Graph the ellipses . 2. Graph the hyperbola . 3. Find the vertices, co-vertices, and asymptotes of , then graph. 4. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0).

  11. vertices: (–6, ±5); co-vertices (6, 0), (–18, 0); 5 asymptotes: y = ± (x + 6) 12 Notes 3. Find the vertices, co-vertices, and asymptotes of , then graph.

  12. Notes: 4. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0).

  13. The values a, b, and c, are related by the equation c2 = a2 + b2. Also note that the length of the trans-verse axis is 2a and the length of the conjugate is 2b.

  14. As with circles and ellipses, hyperbolas do not have to be centered at the origin.

  15. Hyperbolas: Extra Info The following power-point slides contain extra examples and information. Reminder: Lesson Objectives Write the standard equation for a hyperbola. Graph a hyperbola, and identify its center, vertices, co-vertices, foci, and asymptotes.

  16. Notice that as the parameters change, the graph of the hyperbola is transformed.

  17. Step 1 The equation is in the form so the transverse axis is horizontal with center (0, 0). x2 y2 x2 y2 a2 b2 16 36 – = 1 – = 1 Check It Out! Example 3a: Graphing Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.

  18. Step 3 The equations of the asymptotes are y = x and y = – x . 3 3 2 2 Check It Out! Example 3a Continued Step 2 Because a = 4 and b = 6, the vertices are (4, 0) and (–4, 0) and the co-vertices are (0, 6) and . (0, –6).

  19. Check It Out! Example 3 Step 4 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the box. Step 5 Draw the hyperbola by using the vertices and the asymptotes.

  20. (y + 5)2 (x –1)2 (y – k)2 (x – h)2 1 9 a2 b2 – = 1 – = 1 Check It Out! Example 3b: Graphing Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph. Step 1 The equation is in the form so the transverse axis is vertical with center (1, –5).

  21. Step 3 The equations of the asymptotes are y + 5 = (x –1) and y + 5 = – (x –3). 1 1 3 3 Check It Out! Example 3b Continued Step 2 Because a = 1 and b =3, the vertices are (1, –4) and (1, –6) and the co-vertices are (4, –5) and (–2, –5).

  22. Check It Out! Continued Step 4 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the box. Step 5 Draw the hyperbola by using the vertices and the asymptotes.

  23. Step 1 Because the vertex is on the vertical axis, the transverse axis is vertical and the equation is in the form . Because a = 9 and b = 7, the equation of the graph is , or . y2 x2 y2 y2 x2 x2 81 a2 b2 72 49 92 – = 1 – = 1 – = 1 Check It Out! Example 2a: Writing Equations Write an equation in standard form for each hyperbola. Vertex (0, 9), co-vertex (7, 0) Step 2 a = 9 and b = 7. Step 3 Write the equation.

  24. Step 1 Because the vertex and the focus are on the horizontal axis, the transverse axis is horizontal and the equation is in the form x2 y2 . a2 b2 – = 1 Check It Out! Example 2b: Writing Equations Write an equation in standard form for each hyperbola. Vertex (8, 0), focus (10, 0)

  25. Step 3 The equation of the hyperbola is . x2 y2 64 36 – = 1 Check It Out! Example 2b Continued Step 2 a = 8 and c = 10; Use c2 = a2 + b2to solve for b2. 102 = 82 + b2 Substitute 10 for c, and 8 for a. 36 = b2

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