Parts of a Hyperbola
A hyperbola is the set of points such that the difference of the distances from two fixed points called the foci remains the same. A hyperbola is formed when a plane intersects, or crosses, both halves of a double cone without passing through its apex. The graph of a hyperbola has two unconnected branches that are mirror images of each other.
The properties of a hyperbola include:
- A hyperbola has two axes of symmetry. The transverse axis is the line that passes through the foci of a hyperbola. The conjugate axis is the line perpendicular to the transverse axis that intersects it at the center of the hyperbola.
- A vertex of a hyperbola is one of two points where the transverse axis intersects the hyperbola.
- A hyperbola has two asymptotes that cross at the center of the hyperbola. An asymptote is a line that a graph approaches as one of the variables approaches infinity or negative infinity.
- The fundamental rectangle is a rectangle used as a guide to graph a hyperbola. The vertices of a hyperbola are the midpoints of two sides of the fundamental rectangle.
- The diagonals of the fundamental rectangle, or the segments connecting opposite vertices, lie on the asymptotes of the hyperbola.
Equation of a Hyperbola
Horizontal Transverse Axis Centered at the Origin | Vertical Transverse Axis Centered at the Origin |
The equation is:
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ |
The equation is:
$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ |
The coordinates of the vertices are $(-a,0)$ and $(a,0)$. | The coordinates of the vertices are $(0, -a)$ and $(0,a)$. |
The coordinates of the foci are $(c,0)$ and $(-c,0)$, where:
$a^2+b^2=c^2$ |
The coordinates of the foci are $(0,c)$ and $(0,-c)$ , where:
$a^2+b^2=c^2$ |
The standard form for an equation of a hyperbola with center $(h, k)$ is written using $(x-h)$ in place of $x$ and $(y-k)$ in place of $y$.
- Horizontal transverse axis:
- Vertical transverse axis:
Graphing Hyperbolas
Hyperbolas in the Coordinate Plane
Horizontal Transverse Axis | Vertical Transverse Axis | |
---|---|---|
Equation |
$\frac{x^2}{25}-\frac{y^2}{9}=1$ |
$\frac{y^2}{4}-\frac{x^2}{16}=1$ |
Center |
$(0, 0)$ |
$(0, 0)$ |
Vertices | $(-5, 0)$ and $(5, 0)$ | $(0, -2)$ and $(0, 2)$ |
Transverse axis |
$y=0$ |
$x=0$ |
Conjugate axis |
$x=0$ |
$y=0$ |
Graph |
To graph a hyperbola given its equation in standard form:
1. Use the equation to identify the center of the hyperbola.
2. Use the values of $a$ and $b$ to draw the fundamental rectangle. The length of the rectangle corresponds to the distance between the vertices and is equal to $2a$, and the width of the rectangle is equal to $2b$.
3. Draw diagonal lines through opposite corners of the fundamental rectangle to show the asymptotes.
4. Locate the vertices.
5. Plot additional points on the hyperbola as needed, and sketch two curves through the vertices that approach the asymptotes.
As with circles and ellipses, sometimes, the equation of a hyperbola may need to be rewritten in standard form by completing the square.
The transverse axis is the vertical line $x=h$, or $x=3$.
The conjugate axis is the horizontal line $y=k$, or $y=2$.
Determine the center of the hyperbola and the lengths of $a$ and $b$.
Since $h=3$ and $k=2$, the center is $(3,2)$.
Calculate the vertices. The vertices are both 3 units from the center. The transverse axis is vertical, so the vertices are above and below the center.
Vertices:Identify the foci.
To locate the foci, use the equation:By Hand: Plot the vertices and foci. Draw a smooth curve through the vertex $(3,5)$ that approaches both asymptotes above the fundamental rectangle. Draw another smooth curve through the vertex $(3,-1)$ that approaches both asymptotes below the fundamental rectangle.
On a Graphing Calculator: Solve for $y$.Complete the square for each variable, adding the value to both sides of the equation. Then, factor the expressions on the left side into perfect squares and simplify the right side to write the equation in standard form.
When adding to both sides, remember to account for the coefficient by multiplying the amount added to the $y$-terms by –4.
The transverse axis is the horizontal line $y=k$, or $y=5$.
The conjugate axis is the vertical line $x=h$, or $x=2$.
The vertices are at $(0,5)$ and $(4,5)$.
- Plot the center and the vertices, and draw the transverse and conjugate axes.
- Use the values of $a$ and $b$ to draw the fundamental rectangle.
- Draw the asymptotes through the diagonals of the fundamental rectangle.
- Then sketch the two branches of the hyperbola approaching the asymptotes.