WarmUp Exercises

1
Warm-Up Exercises

• Consider the equation .
• Find the x and y-intercepts of the graph of the
  equation.
• Find the symmetry of the graph of the equation.
• Solve the equation for y.
• Write the equation that would translate the
  equation 2 units to the left and 3 units up.

2
Chapter 9 Topics in Analytical Geometry

• 9-3 Hyperbolas

3
Objective

• To find equations of hyperbolas.
• To graph hyperbolas.

4
Introduction

• The hyperbola is the next Conic Sections we will
  study. Its equation is very similar to that of
  an ellipse studied last section. The terminology
  is also very similar. This

will aid in learning about the new conic
section. A hyperbola is formed when the cone is
sliced with a plane that is parallel to the
center axis of the conic section.
5
Hyperbola Notation

• A hyperbola is the set of all points in
  a plane where the difference of the distances
  from two distinct points, called the foci, to the
  point are constant.

6
Hyperbola Notation

• The line through the two foci intersects the
  hyperbola at two points called vertices. The
  line segment connecting the vertices is called
  the transverse axis, and its midpoint is the
  center of the hyperbola.

7
Standard Equation

• The standard form of the equation of an
  hyperbola, with center is
• Transverse axis is horizontal
• Transverse axis is vertical

8
Standard Equation

• The vertices are a units from the center, and the
  foci are c units from the center, where
  . If the center is at the origin ,
  the equation takes one of the following forms.
• Transverse axis is horizontal
• Transverse axis is vertical

9
Standard Equation

• Transverse axis is horizontal
• Transverse axis is vertical

10
Hyperbolas

• Example
• Find the standard form of the equation of the
  hyperbola with foci and and
  vertices and .

11
Asymptotes of a Hyperbola

• Each hyperbola has two asymptotes that intersect
  at the center of the hyperbola. They pass
  through the vertices of a rectangle with
  dimensions 2a by 2b and center at . The
  line segment of length 2b joining
  and or
• and is the conjugate axis of
  the hyperbola.
• Transverse axis is horizontal.
• Transverse axis is vertical.

12
Hyperbolas

• Example
• Sketch the hyperbola whose equation is
• Sketch the hyperbola given by
• and find the equations of the asymptotes.

13
Hyperbolas

• Example
• Find the standard form of the equation of the
  hyperbola having vertices and
• and having asymptotes
• and

14
Assignment

• In Class Assignment
• P. 234 Class Exercises 1-3
• Homework Assignment
• P. 235 Written Exercises 1-21 odd
• Homework Assignment (Day 2)
• P. 237 Written Exercises 39-44