Conic Sections: The Hyperbola Lesson

In the last lesson, we learned how to graph an ellipse. In this lesson, we will learn how to graph a hyperbola that is centered at the origin. A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points is constant. The equation for the hyperbola looks similar to that of an ellipse. The difference is the minus sign in the middle.
When we work with a hyperbola that is centered at the origin, we will see two different equations.

A Hyperbola with x-intercepts (a,0) and (-a,0)

$$\frac{x^2}{a^2}- \frac{y^2}{b^2}=1$$ Our x-intercepts will occur at:
(0,a) and (0,-a)
This type of hyperbola opens left and right. We will not have any y-intercepts. To think about why let's plug a 0 in for x and solve for y: $$\frac{(0)^2}{a^2}- \frac{y^2}{b^2}=1$$ $$- \frac{y^2}{b^2}=1$$ $$-y^2=b^2$$ To solve for y, we need to take the square root of each side: $$\sqrt{-y^2}=\pm \sqrt{b^2}$$ We can split this problem up into: $$\sqrt{-1}\cdot \sqrt{y^2}=\pm \sqrt{b^2}$$ We know this problem will not have a real solution, therefore, we will not have any y-intercepts.

A Hyperbola with y-intercepts (0,b) and (0,-b)

$$\frac{y^2}{b^2}- \frac{x^2}{a^2}=1$$ Our y-intercepts will occur at:
(b,0) and (-b,0)
This type of hyperbola opens up and down. We will not have any x-intercepts. To think about why let's plug a 0 in for y and solve for x: $$\frac{(0)^2}{b^2}- \frac{x^2}{a^2}=1$$ $$- \frac{x^2}{a^2}=1$$ $$-x^2=a^2$$ To solve for x, we need to take the square root of each side: $$\sqrt{-x^2}=\pm \sqrt{a^2}$$ We can split this problem up into: $$\sqrt{-1}\cdot \sqrt{x^2}=\pm \sqrt{a^2}$$ Again, we know this problem will not have a real solution, therefore, we will not have any x-intercepts.
To graph a hyperbola:

• Locate and plot the intercepts
• Find the fundamental rectangle
  • Endpoints: (a,b),(-a,b),(-a,-b),(a,-b)
• Sketch the asymptotes
  • These are extended from the fundamental rectangle
  • y = (b/a)x
  • y = -(b/a)x
• Sketch the graph: each branch goes through an intercept and approaches but doesn't touch the asymptote

Let's look at an example.
Example 1: Sketch the graph of each hyperbola $$\frac{y^2}{4}- \frac{x^2}{9}=1$$ After inspecting the equation, we can determine this is the graph of a hyperbola that is centered at the origin (0,0) and opens up and down. This means we will have y-intercepts and no x-intercepts.
To graph our hyperbola, we can follow our above steps. We will begin by plotting the intercepts. To find these, take the principal and negative square root of the number underneath y². This will give us 2 and -2, therefore, our y-intercepts will occur at: (0,2) and (0,-2) Now we will find the fundamental rectangle. The endpoints occur at:
(3,2),(-3,2),(-3,-2),(3,-2) Now we can sketch the asymptotes. These have the following equations: $$y=\frac{2}{3}x$$ $$y=-\frac{2}{3}x$$ Lastly, we can sketch the graph of the hyperbola. One branch opens up and the other opens down. Each branch will go through the y-intercept and approach but not touch the asymptotes.