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Hyperbolas in Pre-Calculus: Lesson 6.4

Learn about hyperbolas formed by cutting a rubber band stretched around axes endpoints. Understand equations, foci, and asymptotes of hyperbolas.

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Hyperbolas in Pre-Calculus: Lesson 6.4

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  1. Pre-Calc Hyper---bolas Lesson 6.4 Take an ellipse: This ellipse is formed by a rubber band that is stretched so tightly around the four endpoints at the ends of the major and minor axes; now take a pair of scissors and simultaneously ‘snip’ the rubber band at the endpoints of, say, its minor axis, guess what type of figure is formed?

  2. You guessed it!!!! --- A Hyper—bola (Not a Super bola, or a Rosa Bola, But a Hyper—bola! The equations of hyperbolas, with center at the origin look somewhat familiar: x2 - y2 = 1 or y2 - x2 = 1 a2 b2 a2 b2 Do ‘Hyperbolic’ curves cross both axes? By looking at the equations, can you guess which format would describe the hyperbola graphed above?

  3. The axis your hyperbola crosses is called the ‘major’ axis. Technically, there is no minor axis. In ellipses, the larger of the two numbers we assigned as ‘a2’, but now ‘size’ does not matter. What matters now is ‘which’ number appears under the + variable; x2 or y2 So generally, a2 is always in the first term. The other number is b2 and we do need that information. We will show you why later. Hyperbolas also have ‘focus’ points -- ‘c’ values. To find these we use our old friend: a2 + b2 = c2

  4. The order of our focus points will be the same as for an ellipse—always must be on the ‘major axis’ So if major axis is Horizontal---(+ c, 0) If major axis is vertical -- (0, + c) Last, but not least, all hyperbolas have very special lines which helps define how the curves lie—called ‘asymptotes’. (Remember. These are just lines -- (y = mx + b) so try to remember our work with lines done earlier this year.) We will better go over these when we do a few examples!

  5. Example 1: • Graph the hyperbola: • x2 - y2 = 1 • 36 9 • Find the coordinates of • its ‘foci’. • c. Find equations of its asymptotes

  6. Example 2: a. Graph the hyperbola: (y-2)2 - (x+3)2 = 1 9 36 b. Find the coordinates of its ‘foci’. c. Find equations of its asymptotes

  7. Example 3 • a. Sketch the hyperbola: • x2 – 9y2 + 2x + 36y – 44 = 0 • Find the coordinates of its • vertices and foci • c. Find the equations of its asymptotes.

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