Conics: Standard Form of a Hyperbola

As we've been covering conics, we have learned about ellipses, the standard form of an ellipse, and the standard form of circles, but now let's learn about the standard form of a hyperbola!

##Standard Form of a Hyperbola

We already know what a hyperbola looks like - as a review it is essentially two parabolas facing each other.

The general equation of a hyperbola opening left and right is:

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Where...

$(h, k)$ is the center

and the vertices are $(h \pm a, k)$.

The general equation of a hyperbola opening up and down is:

$$\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1$$

Where...

$(h, k)$ is the center

and the vertices are $(h , k\pm b)$.

To solve for the standard form of the hyperbola, learn about vertices, transverse axes, conjugate axes, and foci of hyperbola next.

You can complete the square to convert a hyperbola from general from to standard form.

Standard Form of a Hyperbola

A hyperbola is a conic section made of two separate curves called the branches of the hyperbola. These branches come from a special set of points where the difference of the distances to two fixed points stays constant.

To study and graph hyperbolas, it is important to understand the equation in standard form. Depending on which direction a hyperbola opens, there are two different equations in standard form that we use.

A hyperbola centered at the origin can have two possible forms, seen above. The form of the equation helps you to determine whether the transverse axis is horizontal or vertical.

• If the $x^2$ term is positive first, the hyperbola opens left and right.
• If the $y^2$ term is positive first, the hyperbola opens up and down.

This trick will help you to graph the hyperbola!

#####The Center

The center of the hyperbola is the midpoint between the two branches. If the hyperbola is centered at the origin, the center is $(0,0)$.

You'll learn about translated hyperbolas in later lessons.

####Transverse Axis and Conjugate Axis

The transverse axis passes through the center and the vertices of the hyperbola. It points in the direction the hyperbola opens.

The conjugate axis is perpendicular to the transverse axis.

####The Vertices

The vertices are located $a$ units from the center along the transverse axis.

• For a horizontal hyperbola centered at $(h,k)$: $(h \pm a, k)$
• For a vertical hyperbola centered at $(h,k)$: $(h, k \pm a)$

These are the coordinates for the vertices.

####The Foci

The foci are farther from the center than the vertices. To find the coordinates of its foci, you can use the following formula:

$$c^2 = a^2 + b^2$$

This is different from ellipses, where subtraction is used.

So, we can say that:

• Horizontal hyperbolas: $(h \pm c, k)$
• Vertical hyperbolas: $(h, k \pm c)$

You can use the distance formulas to find missing values if needed. You can also use the foci to graph.

####Asymptotes

The asymptotes of the hyperbola are dashed guide lines that the branches approach but never touch. The hyperbola gets closer and closer to these lines as it moves away from the center.

For a hyperbola centered at $(h,k)$, the equations of the asymptotes are:

• For the horizontal transverse axis: $y-k = \pm \frac{b}{a}(x-h)$
• For the vertical transverse axis: $y-k = \pm \frac{a}{b}(x-h)$

Before drawing the graph of the hyperbola, it is helpful to sketch the asymptotes first.

Once you've practiced working with and graphing hyperbolas, you will be able to solve word problems.

Standard Form of a Hyperbola

by Anusha Rahman

In this lesson by Anusha Rahman, you will learn about the standard form of a hyperbola equation.

Summary

A hyperbola is a type of conic section formed by a plane intersecting a double cone.

Depending on whether the major axis of the hyperbola is along the $x$-axis or $y$-axis, the equation can take one of two forms:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} \text{ OR } \frac{y^2}{b^2} - \frac{x^2}{a^2}$$