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?

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 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

(0,b) (0,-b) Vertex (a,0) Vertex (-a,0) Asymptotes This is an example of a horizontal transverse axis (a, the biggest number, is under the x 2 term with the minus before the y) Focus

Vertical transverse axis

Standard Form of Hyperbola w/ origin Equation Transvers e Axis AsymptotesVertices Horizontaly=+/- (b/a)x(+/-a,o) Verticaly=+/- (a/b)x(0,+/-a) Foci lie on transverse axis, c units from the center c 2 = a 2 +b 2

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

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 origin because focus & vertices are equidistant from the origin Since c=3 & a=2, c 2 = b 2 + a 2 9 = b = b 2 +/-√5 = b

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? c 2 = a 2 + b 2 How do you graph a hyperbola? Plot a and b, draw a box with diagonals. Draw the hyperbola following the diagonals through the vertices.

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 c 2 = a 2 + b 2 ) and substitute into the equation.

Assignment Page 619, 15-18, odd, odd, skip 37

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

Write the equation of the hyperbola in standard form. 16y 2 −36x = 0 16y 2 −36x 2 = −9

Translated Hyperbolas In the following equations the point (h,k) is the center of the hyperbola. Horizontal axis Vertical axis Remember c 2 = a 2 + b 2

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:

Center (0,5) (h,k) a = 4, c = 6 c 2 = a 2 + b = b 2 36 = 16 + b 2 20 = b 2 Center? (5,4) (5,−4) (5,6) ( 5,−6)

Graphing the Equation of a Translated Hyperbola Graph (y + 1) 2 – = 1. (x + 1) 2 4 S OLUTION The y 2 -term is positive, so the transverse axis is vertical. Since a 2 = 1 and b 2 = 4, you know that a = 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. (–1, –2) (–1, 0) (–1, –1)

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

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

Assignment Page 618, even even Page 628, 19-20, 23, 26