# Conics

## Hyperbolas


### Graph a Hyperbola with Center at (0, 0)

The last conic section we will look at is called a hyperbola. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.
We define a hyperbola as all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci, is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Again our goal is to connect the geometry of a conic with algebra. Placing the hyperbola on a rectangular coordinate system gives us that opportunity. In the figure, we placed the hyperbola so the foci  are on the x-axis and the center is the origin.
The definition states the difference of the distance from the foci to a point  is constant. So  is a constant that we will call  so  We will use the distance formula to lead us to an algebraic formula for an ellipse.

To graph the hyperbola, it will be helpful to know about the intercepts. We will find the x-intercepts and y-intercepts using the formula.

The a, b values in the equation also help us find the asymptotes of the hyperbola. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the x, y values get larger and larger.
To find the asymptotes, we sketch a rectangle whose sides intersect the x-axis at the vertices   and intersect the y-axis at   The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola.
The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. They have equations  and 
There are two equations for hyperbolas, depending whether the transverse axis is vertical or horizontal. We can tell whether the transverse axis is horizontal by looking at the equation. When the equation is in standard form, if the x2-term is positive, the transverse axis is horizontal. When the equation is in standard form, if the y2-term is positive, the transverse axis is vertical.
The second equations could be derived similarly to what we have done. We will summarize the results here.
We will use these properties to graph hyperbolas.
We summarize the steps for reference.
Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it.

### Graph a Hyperbola with Center at 

Hyperbolas are not always centered at the origin. When a hyperbola is centered at  the equations changes a bit as reflected in the table.
We summarize the steps for easy reference.
Be careful as you identify the center. The standard equation has  and  with the center as 
Again, sometimes we have to put the equation in standard form as our first step.

### Identify Conic Sections by their Equations

Now that we have completed our study of the conic sections, we will take a look at the different equations and recognize some ways to identify a conic by its equation. When we are given an equation to graph, it is helpful to identify the conic so we know what next steps to take.
To identify a conic from its equation, it is easier if we put the variable terms on one side of the equation and the constants on the other.

### Key Concepts

1. Hyperbola: A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant.


Each of the fixed points is called a focus of the hyperbola.
The line through the foci, is called the transverse axis.
The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola.
The midpoint of the segment joining the foci is called the center of the hyperbola.
The line perpendicular to the transverse axis that passes through the center is called the conjugate axis.
Each piece of the graph is called a branch of the hyperbola.




2. How to graph a hyperbola centered at 




3. How to graph a hyperbola centered at 





### Practice Makes Perfect

Graph a Hyperbola with Center at 
In the following exercises, graph.
Graph a Hyperbola with Center at 
In the following exercises, graph.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.
Mixed Practice
In the following exercises, graph each equation.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?