CONIC SECTIONS THE HYPERBOLA Colleen Beaudoin February 2009

CONIC SECTIONS: THE HYPERBOLA Colleen Beaudoin February, 2009

Hyperbola � Review: The geometric definition relies on a cone and a plane intersecting it � Algebraic definition: a set of points in the plane such that the difference of the distances from two fixed points, called foci, remains constant.

Hyperbola From each point in the plane, the difference of the distances to the foci is a constant. Example: A d 1 d 2 f 1 d 2 d 1 foci B Point A: d 1 -d 2 = c Point B: d 1 -d 2 = c

Items referenced on the graph of a hyperbola Center Transverse Axis f 1 f 2 Conjugate Axis foci Vertices

Hyperbola Vs. Ellipse Algebraic Definition of a hyperbola: a set of points in the plane such that the difference of the distances from two fixed points, called foci, remains constant. Ø Ø How is the definition similar to that of an ellipse? How is it different?

Facts: Hyperbola Equation q q q Both variables are squared. Equation: Compare the equations of ellipses and hyperbolas. What makes the hyperbola different from the parabola? What makes the hyperbola different from a circle?

where the center is at (h, k) and |2 a | is the length of the horizontal axis and |2 b| is the of the length of the vertical axis. Procedure to graph: 1. Put in standard form (above): x squared term - y squared term = 1 2. Determine if the hyperbola is opening vertically or horizontally. (If x is first, it’s horizontal. If y is first, it’s vertical. ) 3. Plot the center (h, k) 4. Plot the endpoints of the horizontal axis by moving “a” units left and right from the center.

To graph: 5. Plot the endpoints of the vertical axis by moving “b” units up and down from the center. Note: The line segment that contains the vertices of the hyperbola is known as the transverse axis. The other axis is the conjugate axis. 6. Draw a rectangle such that each of the axis endpoints is the midpoint of a side.

To graph: 7. Sketch the diagonals of the rectangle and extend them outside of the rectangle. (These are the asymptotes of the hyperbola. ) 8. Draw each branch of the hyperbola – Be sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.

To graph: 9. Use the following formula to help locate the foci: c 2 = a 2 + b 2 Move “c” units left and right form the center if the transverse axis is horizontal OR Move “c” units up and down form the center if the transverse axis is vertical Label the points f 1 and f 2 for the two foci. Note: It is not necessary to plot the foci to graph the hyperbola, but it is common practice to locate them.

The equation of each asymptote can be found by using the point-slope formula. Use the center as “the point” and slope can be found by counting on the graph (from the point to the corner of the rectangle). Or the following formulas can be used:

Exp. 1: Graph 1. Put in standard form. Done 2. Determine if the hyperbola is opening vertically or horizontally. Vertically because “y” is first. 3. Identify the center. (0, 0) 4. Identify the endpoints of the horizontal axis. (6, 0) and (-6, 0) 5. Identify the endpoints of the vertical axis. (0, 8) and (0, -8) Which pair of endpoints are the vertices? (0, 8) and (0, -8)

Exp. 1: Graph 6. Draw a rectangle such that each of the axis endpoints is the midpoint of a side. 7. Sketch the asymptotes of the hyperbola. 8. Draw each branch of the hyperbola – Be sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.

Exp. 1: Graph 9. Locate the foci. (0, 10) and (0, -10) 10. Find the equations of the asymptotes.

Transverse axis Conjugate axis Center Asymptotes

Exp. 2: Graph 1. Put in standard form. Done 2. Determine if the hyperbola is opening vertically or horizontally. Horizontally because “x” is first. 3. Identify the center. (3, -2) 4. Identify the endpoints of the horizontal axis. (-3, -2) and (9, -2) 5. Identify the endpoints of the vertical axis. (3, 2) and (3, -6) Which pair of endpoints are the vertices? (-3, -2) and (9, -2)

Exp. 2: Graph 6. Draw a rectangle such that each of the axis endpoints is the midpoint of a side. 7. Sketch the asymptotes of the hyperbola. 8. Draw each branch of the hyperbola – Be sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.

Exp. 2: Graph 9. Locate the foci. (3+2√ 13, -2) and (3 -2√ 13, -2) 10. Find the equations of the asymptotes.

Exp. 3: Write the equation in standard form and graph: 1. Put in standard form.

Challenge Question Write the equation for a hyperbola with xintercepts at 5 and -5 and foci (6, 0) and (-6, 0).

Review 1) 2) 3) 4) 5) 6) How can you tell if the graph of an equation will be a line, parabola, circle, ellipse, or hyperbola? What’s the standard form of an ellipse? What’s the standard form of a parabola? What’s the standard form of a circle? How are the various equations similar and different?