9.3 Hyperbolas Hyperbola: set of all points such that the difference of the distances from any point to the foci is constant.

(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 The transverse axis is the line segment joining the vertices(through the foci) The midpoint of the transverse axis is the center of the hyperbola..

Standard Equation of a Hyperbola (Center at Origin) This is the equation if the transverse axis is horizontal. The foci of the hyperbola lie on the major axis (the y-axis, or the horizontal), c units from the center, where c 2 = a 2 + b 2 (c, 0) (–c, 0) (–a, 0)(a, 0) (0, b) (0, –b)

The standard form of the Hyperbola with a center at (h, k) and a horizontal axis is……

The Hyperbola with a center at (h, k) and a horizontal axis has the following characteristics……

Standard Equation of a Hyperbola (Center at Origin) This is the equation if the transverse axis is vertical. The foci of the hyperbola lie on the major axis (the x-axis or the vertical), c units from the center, where c 2 = a 2 + b 2 (0, c) (0, –c) (0, –a) (0, a) (b, 0)(–b, 0)

The standard form of the Hyperbola with a center at (h, k) and a vertical axis is…… Note the h and k!!!

The Hyperbola with a center at (h, k) and a vertical axis has the following characteristics……

Hyperbola – General Rules… -x and y are both squared -Equation always equals 1 -Equation is always minus(-) -a 2 is always the first denominator -c 2 = a 2 + b 2 -c is the distance from the center to each foci on the transverse axis -a is the distance from the center to each vertex on the transverse axis

General Rules Continued… -b is the distance from the center to each midpoint of the rectangle used to draw the asymptotes. This distance runs perpendicular to the distance (a). -Transverse axis has a length of 2a. -If x 2 is first then the hyperbola is horizontal. -If y 2 is first then the hyperbola is vertical.

Still More Rules… -The center is in the middle of the 2 vertices and the 2 foci. -The vertices and the co-vertices are used to draw the rectangles that form the asymptotes. -The vertices and the co-vertices are the midpoints of the rectangle. -The co-vertices are not labeled on the hyperbola because they are not actually part of the graph.

Ex 1: Write the equation in standard form of 4x 2 – 16y 2 = 64. Find the foci and vertices of the hyperbola. Get the equation in standard form (make it equal to 1): 4x 2 – 16y 2 = Use c 2 = a 2 + b 2 to find c. c 2 = c 2 = = 20 c = (c, 0)(–c,0) (–4,0)(4, 0) (0, 2) (0,-2) That means a = 4 b = 2 Vertices: Foci: Simplify... x 2 – y 2 =

Ex 2: Write an equation of the hyperbola whose foci are (0, –6) and (0, 6) and whose vertices are (0, –4) and (0, 4). Its center is (0, 0). y 2 – x 2 = 1 a 2 b 2 Since the major axis is vertical, the equation is the following: Since a = 4 and c = 6, find b... c 2 = a 2 + b = b 2 36 = 16 + b 2 20 = b 2 The equation of the hyperbola: y 2 – x 2 = (–b, 0)(b, 0) (0, 4) (0, –4) (0, 6) (0, –6)

How do you graph a hyperbola? To graph a hyperbola, you need to know the center, the vertices, the co-vertices, and the asymptotes... Draw a rectangle using +a and +b as the sides... (5, 0)(–5,0) (–4,0)(4, 0) (0, 3) (0,-3) a = 4 b = 3 The asymptotes intersect at the center of the hyperbola and pass through the corners of a rectangle with corners (+ a, + b) Ex 3: Graph the hyperbola x 2 – y 2 = c = 5 Draw the asymptotes (diagonals of rectangle)... Draw the hyperbola... Here are the equations of the asymptotes: Horizontal Transverse Axis: y = + b x a Vertical Transverse Axis: y = + a x b

Ex 4: 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.

Examples