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

3. Find the vertices, co-vertices, and asymptotes of , then graph. Notes A. x2 4 + = 1 y2 9 1. Graph the ellipses . B. (x-3)2 16 + = 1 (y+1)2 25 2. Graph the hyperbola . (x-3)2 16 - = 1 (y+1)2 25 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). 2

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.

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

Example 1 Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph x2 49 – = 1 y2 9 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.

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

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.

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

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 1 Because the vertex and the focus are on the horizontal axis x2 16 – = 1 y2 b2 Step 2 Use focus2 = 16 + other denominator. Step 3 The equation of the hyperbola is . x2 16 – = 1 y2 84

3. Find the vertices, co-vertices, and asymptotes of , then graph. Notes A. x2 4 + = 1 y2 9 1. Graph the ellipses . B. (x-3)2 144 + = 1 (y+1)2 25 2. Graph the hyperbola . (x-3)2 144 - = 1 (y+1)2 25 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). 10

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

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

The values a, b, and c, are related by the equation c2 = a2 + b2 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.

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

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. 15

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

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

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). Step 3 The equations of the asymptotes are y = x and y = – x . 3 2

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.

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

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). Step 3 The equations of the asymptotes are y + 5 = (x – 1) and y + 5 = – (x – 3). 1 3

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.

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

Check It Out! Example 2b: Writing Equations Write an equation in standard form for each hyperbola. Vertex (8, 0), focus (10, 0) x2 a2 – = 1 y2 b2 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 .

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