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10.5 Hyperbolas

10.5 Hyperbolas. p.615 What are the parts of a hyperbola? What are the standard form equations of a hyperbola? How do you know which way it opens? Given a & b, how do you find the value of c? How do you graph a hyperbola? Why does drawing a box make graphing easier?

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10.5 Hyperbolas

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  1. 10.5 Hyperbolas p.615 What are the parts of a hyperbola? What are the standard form equations of a hyperbola? How do you know which way it opens? Given a & b, how do you find the value of c? How do you graph a hyperbola? Why does drawing a box make graphing easier? How do you write the equation from a graph?

  2. Hyperbolas • Like an ellipse but instead of the sum of distances it is the difference • A hyperbola is the set of all points P such that the differences from P to two fixed points, called foci, is constant • The line thru the foci intersects the hyperbola @ two points (the vertices) • The line segment joining the vertices is the transverse axis, and it’s midpoint is the center of the hyperbola. • Has 2 branches and 2 asymptotes • The asymptotes contain the diagonals of a rectangle centered at the hyperbolas center

  3. Asymptotes (0,b) Vertex (a,0) Vertex (-a,0) Focus Focus (0,-b) This is an example of a horizontal transverse axis (a, the biggest number, is under the x2 term with the minus before the y)

  4. Vertical transverse axis

  5. Standard Form of Hyperbola w/ center @ origin Foci lie on transverse axis, c units from the center c2 = a2+b2

  6. Graph 4x2 – 9y2 = 36 • Write in standard form (divide through by 36) • a=3 b=2 – because x2 term is ‘+’ transverse axis is horizontal & vertices are (-3,0) & (3,0) • Draw a rectangle centered at the origin. • Draw asymptotes. • Draw hyperbola.

  7. Write the equation of a hyperbole with foci (0,-3) & (0,3) and vertices (0,-2) & (0,2). • Vertical because foci & vertices lie on the y-axis • Center @ origin because focus & vertices are equidistant from the origin • Since c=3 & a=2, c2 = b2 + a2 • 9 = b2 + 4 • 5 = b2 • +/-√5 = b

  8. What are the parts of a hyperbola? Vertices, foci, center, transverse axis & asymptotes • What are the standard form equations of a hyperbola? • How do you know which way it opens? Transverse axis is always over a • Given a & b, how do you find the value of c? c2 = a2 + b2 • How do you graph a hyperbola? Plot a and b, draw a box with diagonals. Draw the hyperbola following the diagonals through the vertices.

  9. Why does drawing a box make graphing easier? The diagonals of the box are the asymptotes of the hyperbola. • How do you write the equation from a graph? Identify the transverse axis, find the value of a and b (may have to use c2 = a2 + b2) and substitute into the equation.

  10. Assignment Page 619, 15-18, 19-39 odd, 57-61 odd, skip 37

  11. 10.5 Hyperbolas, day 2 What are the standard form equations of a hyperbola if the center has been translated?

  12. Write the equation of the hyperbola in standard form. 16y2 −36x2 + 9 = 0 16y2 −36x2 = −9

  13. Translated Hyperbolas In the following equations the point (h,k) is the center of the hyperbola. Horizontal axis Vertical axis Remember c2 = a2 + b2

  14. Write an equation for the hyperbola. Vertices at (5, −4) and (5,4) and foci at (5,−6) and (5,6). Draw a quick graph. Equation will be:

  15. (5,6) (5,4) Center (0,5) (h,k) a = 4, c = 6 c2 = a2 + b2 62 = 42 + b2 36 = 16 + b2 20 = b2 Center? (5,−4) (5,−6)

  16. Graphing the Equation of a Translated Hyperbola (x + 1)2 Graph (y + 1)2– = 1. 4 (–1, 0) (–1, –1) (–1, –2) SOLUTION The y2-term is positive, so thetransverse axis is vertical. Sincea2 = 1 and b2 = 4, you know thata = 1 and b = 2. Plot the center at (h, k) = (–1, –1). Plot the vertices 1 unit above and below the center at (–1, 0) and (–1, –2). Draw a rectangle that is centered at (–1, –1) and is 2a = 2 units high and 2b = 4 units wide.

  17. Graphing the Equation of a Translated Hyperbola (x + 1)2 Graph (y + 1)2– = 1. 4 (–1, 0) (–1, –1) (–1, –2) SOLUTION The y2-term is positive, so thetransverse axis is vertical. Sincea2 = 1 and b2 = 4, you know thata = 1 and b = 2. Draw the asymptotes through the corners of the rectangle. Draw the hyperbola so that it passes through the vertices and approaches the asymptotes.

  18. What are the standard form equations of a hyperbola if the center has been translated?

  19. Assignment Page 618, 20-38 even. 50-58 even Page 628, 19-20, 23, 26

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