Hyperbola – Properties, Components, and Graph

The hyperbola is a unique type of conic section where we see two disjointed curves representing its equation. These conics are used in describing the pathways of a spacecraft and are even used to model certain seismological events.

Hyperbolas are conic sections that are the result of a plane intersecting both surfaces of a double cone. Their graphs are like two U-shaped curves that are facing each other vertically or horizontally.

You may have been introduced to hyperbolas when you first learned about conic sections, so if you need a quick refresher, feel free to take a look at this article to see what makes hyperbolas different from parabolas and ellipses.

In this article, we’ll focus on hyperbolas and learn the following:

  • Understanding how hyperbolas are obtained and the different components it contains.

  • Identify the different standard forms of a hyperbola equation.

  • Know how to represent these conics on the $xy$-coordinate system.

We’ll also try out doing the reverse- finding the equations representing the hyperbolas given their graphs. This article thoroughly covers all components of hyperbolas, so make sure to take notes.

For now, why don’t we begin by refreshing how hyperbolas are formed using a double cone and a plane?

What is a hyperbola?             

Hyperbolas result when a plane and right double cone intersect each other and cover upper and lower intersections, as shown below.

This means that hyperbolas are two U-shaped curves (called branches) that face opposite each other. They can either be oriented vertically as our example or oriented horizontally.

The image above shows us the different components of a hyperbola regardless of their orientation.

  • Hyperbolas have two curves called the branches facing opposite each other.

  • Since they have two U-shaped curves, hyperbolas will also have two vertices and foci.

  • The transverse axis is a guiding axis that divides the hyperbola in half.

Formal hyperbola definition

The components mentioned above help us define hyperbolas formally. Hyperbolas are conic sections where all points that lie on their graphs satisfy the following condition:

  • Let’s say $P(x,y)$ lies on the hyperbola, determine the distances between $P$ and the two foci.

  • When lying on a hyperbola, the difference between these two distances will always be constant.

This means that for hyperbolas like the image shown above, $P_1$ and $P_2$ are two points lying on the hyperbola. The (absolute value) difference between $\overline{P_2F_1}$ and $\overline{P_2F_2}$ will be equal to the difference between $\overline{P_1F_1}$ and $\overline{P_1F_2}$.

All hyperbolas’ variations will satisfy these conditions, and these conditions make hyperbolas unique from the rest of the conic sections.

How to find the equation of a hyperbola?             

Four standard forms are important for us to keep in mind when working with hyperbolas and their equations. The factors affecting the form of the hyperbola’s equation are the following:

  • The center of a hyperbola will affect its equation.

  • The orientation of the graph will affect which term would be placed first.

Hyperbola formula in standard forms

We’ll divide this section into the standard forms of the hyperbola centered at the origin, $(0, 0)$, and centered at the vertex, $(h, k)$.

Standard Form of Hyperbolas Centered at the Origin

When hyperbolas are centered at the origin, we expect no constants inside the squared term. Here’s a table showing the two possible forms of the equation:

Parabola’s Orientation

Equation

Vertical

$\dfrac{y^2}{a^2} – \dfrac{x^2}{b^2} = 1$

Horizontal

$\dfrac{x^2}{a^2} – \dfrac{y^2}{b^2} = 1$

We can see that the leading term gives us an idea of how the hyperbola’s graph will appear – when the leading term is $x^2$, the hyperbolas are symmetric horizontally. Similarly, when $y^2$ is the leading term, the hyperbolas will be symmetric vertically.

Standard Form of Hyperbolas Centered at $\boldsymbol{(h, k)}$

When hyperbolas are not centered at the origin and instead at $(h, k)$.

Parabola’s Orientation

Equation

Vertical

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

Horizontal

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

The only difference between these pair of equations is the fact that the hyperbola is translated $h$ units horizontally and $k$ units vertically. The rest of the hyperbola’s components will also be affected by this translation.

How to find the vertices of a hyperbola?

The vertices of a hyperbola are the two points showing the minimum and maximum values possible for the leading term. Regardless of the center’s location, the vertices’ distance from the center will be dependent on the first denominator.

Equation

Center

Vertex

$\dfrac{y^2}{a^2} – \dfrac{x^2}{b^2} = 1$

$(0,0)$

$(0, -a), (0, a)$

$\dfrac{x^2}{a^2} – \dfrac{y^2}{b^2} = 1$

$(-a, 0), (a, 0)$

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

$(h, k)$

$(h, k-a), (h, k + a)$

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

$(h -a , k), (h + a, k)$

To summarize the table shown above, to find the vertices, we can either:

  • Move $a$ units to both directions of the center when we need the graph of the hyperbola.

  • Add and subtract $a$ from the center’s coordinate – whichever is the leading term, that’s the coordinate we’ll be adding and subtracting $a$ from.

How to find the foci of a hyperbola?

The foci will depend on the values of the denominators from the hyperbola’s equation.

\begin{aligned}c^2 = a^2 + b^2\end{aligned}

This means that we can find the distance of the foci from the center by adding the denominators and then taking the sum’s square root.

Once we have the value of $c$, we move $c$ units away from the center to locate the hyperbola’s two foci.

Equation

Center

Vertex

$\dfrac{y^2}{a^2} – \dfrac{x^2}{b^2} = 1$

$(0,0)$

$(0, -c), (0, c)$

$\dfrac{x^2}{a^2} – \dfrac{y^2}{b^2} = 1$

$(-c, 0), (c, 0)$

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

$(h, k)$

$(h, k-c), (h, k + c)$

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

$(h -c , k), (h + c, k)$

This means that we count $c$ units on both sides of the center and along the direction of the hyperbola to find the foci from the center.

How to find the asymptotes of a hyperbola?

Hyperbolas have a pair of asymptotes that have a general form of $y = mx + b$. This means that we’re looking for two linear equations when finding the asymptotes of a hyperbola.

The asymptotes’ equations will depend on the square roots of the denominators. We can also find the hyperbola equation centered at $(h, k)$ by simply translating it.

Equation

Asymptotes’ Equations

$\dfrac{y^2}{a^2} – \dfrac{x^2}{b^2} = 1$

$y = \dfrac{a}{b}x$

$y = -\dfrac{a}{b}x$

$\dfrac{x^2}{a^2} – \dfrac{y^2}{b^2} = 1$

$y = \dfrac{b}{a}x$

$y = -\dfrac{b}{a}x$

$\dfrac{(y – k)^2}{a^2} – \dfrac{(x