Hyperbolas
By the end of this section, you will be able to:
- Graph a hyperbola with center at
- Graph a hyperbola with center at
- Identify conic sections by their equations
Before you get started, take this readiness quiz.
- Solve:
If you missed this problem, review [link].
- Expand:
If you missed this problem, review [link].
- Graph
If you missed this problem, review [link].
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.
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.
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.
Standard Form of the Equation a Hyperbola with Center
The standard form of the equation of a hyperbola with center
is
Notice that, unlike the equation of an ellipse, the denominator of
is not always
and the denominator of
is not always
Notice that when the
-term is positive, the transverse axis is on the x-axis. When the
-term is positive, the transverse axis is on the y-axis.
Standard Forms of the Equation a Hyperbola with Center |
|
|
|
Orientation |
Transverse axis on the x-axis. Opens left and right |
Transverse axis on the y-axis. Opens up and down |
Vertices |
|
|
x-intercepts |
|
none |
y-intercepts |
none |
|
Rectangle |
Use |
Use |
asymptotes |
|
|
We will use these properties to graph hyperbolas.
How to Graph a Hyperbola with Center
![Step 1 is to write the equation in standard form. The the quantity x squared divided by 25 end quantity minus the quantity y squared divided by 4 end quantity is equal to 1 is already in standard form.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_006a_img.jpg) ![Step 2 is to determine whether the transverse axis is horizontal or vertical. Since the x squared term is positive, the transverse axis is horizontal.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_006b_img.jpg) ![Step 3 is to find the vertices. Since a squared is equal to 25, then a is equal to plus or minus 5. The vertices lie on the x-axis and are (negative 5, 0) and (5, 0).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_006c_img.jpg) ![Step 4 is to sketch the rectangle centered at the origin, intersecting one axis at plus or minus a and the other at plus or minus b. Since a is equal to plus or minus 5, the rectangle will intersect the x-axis at the vertices. Since b is equal to plus or minus 2, the rectangle will intersect the y-axis at (0, negative 2) and (0, 2). The rectangle is shown on a coordinate plane with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_006d_img.jpg) ![Step 5 is to sketch the asymptotes, the lines through the diagonals of the rectangle. The asymptotes have the equations y is equal to five-halves times x and y is equal to negative five-halves x. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled and the lines that represent the asymptotes.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_006e_img.jpg) ![Step 6 is to draw the two branches of the hyperbola. Start at each vertex and use the asymptotes as a guide. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled, the lines that represent the asymptotes, y is equal to plus or minus five-halves times x, and the branches that pass through (plus or minus 5, 0) and open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_006f_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_302_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus four-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_303_img.jpg)
We summarize the steps for reference.
Graph a hyperbola centered at
- Write the equation in standard form.
- Determine whether the transverse axis is horizontal or vertical.
- Find the vertices.
- Sketch the rectangle centered at the origin intersecting one axis at
and the other at
- Sketch the asymptotes—the lines through the diagonals of the rectangle.
- Draw the two branches of the hyperbola.
Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it.
|
|
To write the equation in standard form, divide each term by 64 to make the equation equal to 1. |
|
Simplify. |
|
Since the y2-term is positive, the transverse axis is vertical. Since then |
|
The vertices are on the y-axis, Since then |
|
Sketch the rectangle intersecting the x-axis at and the y-axis at the vertices. Sketch the asymptotes through the diagonals of the rectangle. Draw the two branches of the hyperbola. |
|
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_304_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus three-fifths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_305_img.jpg)
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.
Standard Forms of the Equation a Hyperbola with Center |
|
|
|
Orientation |
Transverse axis is horizontal. Opens left and right |
Transverse axis is vertical. Opens up and down |
Center |
|
|
Vertices |
a units to the left and right of the center |
a units above and below the center |
Rectangle |
Use a units left/right of center b units above/ below the center |
Use a units above/below the center b units left/right of center |
How to Graph a Hyperbola with Center
![Step 1 is to write the equation in standard form. Notice that the equation the quantity x minus 1 squared all divided by 9 end quantity minus the quantity y minus 2 squared all divided by 16 end quantity is equal to 1 is already in standard form.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_008a_img.jpg) ![Step 2 is to deteremine whether the transverse axis is horizonal or vertical. Since the x squared term is positive, the hyperbola opens left and right. The transverse axis is horizontal. The hyperbola opens left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_008b_img.jpg) ![Step 3 is to find the center and a and b. h is equal to 1 and k is equal 2. a squared is equal to 9 and b squared is equal to 16. You can see tha x minus h is x minus 1, and that y minus k is y minus 2. So, the center is (1, 2) and a is equal to 3 and b is equal to 4.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_008c_img.jpg) ![Step 4 is to sketch the rectangle centered at (h, k) using a and b. Mark the center (1, 2) on a coordinate plane. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 4 units above and below the center.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_008d_img.jpg) ![Step 5 is to sketch the asymptotes on the coordinate plane. They are the lines through the diagonals of the retcangle. Mark the vertices which lie on the rectangle 3 units to the left and right of the center. The vertices are (negative 2, 2) and (4, 2).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_008e_img.jpg) ![Step 6 is to draw the branches of the hyperbola. Start the vertices, (negative 2, 2) and (4, 2) and use the asymptotes as a guide. The branches should open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_008f_img.jpg)
Graph
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with an asymptote that passes through (negative 2, negative 2) and (8, 4) and an asymptote that passes through (negative 2, 4) and (8, negative 2), and branches that pass through the vertices (negative 2, 2) and (8, 2) and opens left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_306_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (2, 2), an asymptote that passes through (0, negative 1) and (4, 5) and an asymptote that passes through (0, 5) and (4, negative 1), and branches that pass through the vertices (0, 2) and (4, 2) and opens left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_307_img.jpg)
We summarize the steps for easy reference.
Graph a hyperbola centered at
- Write the equation in standard form.
- Determine whether the transverse axis is horizontal or vertical.
- Find the center and a, b.
- Sketch the rectangle centered at
using a, b.
- Sketch the asymptotes—the lines through the diagonals of the rectangle. Mark the vertices.
- Draw the two branches of the hyperbola.
Be careful as you identify the center. The standard equation has
and
with the center as
|
|
Since the term is positive, the hyperbola opens up and down. |
|
Find the center, |
Center: |
Find a, b. |
|
Sketch the rectangle that goes through the points 3 units above and below the center and 2 units to the left/right of the center. Sketch the asymptotes—the lines through the diagonals of the rectangle. Mark the vertices. Graph the branches. |
|
Graph
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 3), an asymptote that passes through (negative 5, negative 7) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, 7), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 7) and opens up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_308_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 2), an asymptote that passes through (negative 5, negative 5) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, negative 5), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 5) and opens up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_309_img.jpg)
Again, sometimes we have to put the equation in standard form as our first step.
ⓐ Write the equation in standard form and ⓑ graph
ⓐ
ⓑ* * *
ⓐ Write the equation in standard form and ⓑ graph
ⓐ
ⓑ* * *
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.
Conic |
Characteristics of |
and
terms |
Example |
|
{: valign=”top”} |
———- |
|
|
Parabola |
Either |
OR
Only one variable is squared. |
|
|
|
|
{: valign=”top”} |
Circle |
|
and
terms have the same coefficients |
|
|
|
|
{: valign=”top”} |
Ellipse |
|
and
terms have the same sign, different coefficients |
|
|
|
|
{: valign=”top”} |
Hyperbola |
|
and
terms have different signs, different coefficients |
|
|
{: valign=”top”}{: summary=”This table has three columns and five rows. The first row is a header row and it labels each column, “Conic,” “Characteristics of x squared and y squared terms,” and “Example.” The first column is a header column and it labels each row “Parabola,” “Circle,” “Ellipse,”, and “Hyperbola.” In row two, the Parabola is described as having either x squared or y squared and only one variable squared and the example is x is equal to 3 y squared minus 2 y plus 1. In row three, the Circle is described as having x squared and y squared terms with the same coefficients and the example is x squared plus y squared is equal to 49. In row four, the Ellipse is described as having x squared and y squared terms that have the same sign and different coefficients and the example is 4 x squared plus 25 y squared is equal to 100. In row five, the Hyperbola is described as having x squared and y squared terms that have different signs and different coefficients and the example is 25 y squared minus 4 x squared is equal to 100.” .unnumbered data-label=””}
Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.
ⓐ
ⓑ
ⓒ
ⓓ
ⓐ* * *
ⓑ* * *
ⓒ* * *
ⓓ* * *
Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.
ⓐ
ⓑ
ⓒ
ⓓ
ⓐ circle ⓑ ellipse ⓒ parabola ⓓ hyperbola
Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.
ⓐ
ⓑ
ⓒ
ⓓ
ⓐ ellipse ⓑ parabola ⓒ circle ⓓ hyperbola
Key Concepts
- 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.
Standard Forms of the Equation a Hyperbola with Center |
|
|
|
Orientation |
Transverse axis on the x-axis. Opens left and right |
Transverse axis on the y-axis. Opens up and down |
Vertices |
|
|
x-intercepts |
|
none |
y-intercepts |
none |
, |
Rectangle |
Use |
Use |
asymptotes |
|
|
-
**How to graph a hyperbola centered at
**
- Write the equation in standard form.
- Determine whether the transverse axis is horizontal or vertical.
- Find the vertices.
- Sketch the rectangle centered at the origin intersecting one axis at
and the other at
- Sketch the asymptotes—the lines through the diagonals of the rectangle.
- Draw the two branches of the hyperbola.
Standard Forms of the Equation a Hyperbola with Center |
|
|
|
Orientation |
Transverse axis is horizontal. Opens left and right |
Transverse axis is vertical. Opens up and down |
Center |
|
|
Vertices |
a units to the left and right of the center |
a units above and below the center |
Rectangle |
Use a units left/right of center b units above/below the center |
Use a units above/below the center b units left/right of center |
Conic |
Characteristics of |
and
terms |
Example |
|
{: valign=”top”} |
———- |
|
|
Parabola |
Either |
OR
Only one variable is squared. |
|
|
|
|
{: valign=”top”} |
Circle |
|
and
terms have the same coefficients |
|
|
|
|
{: valign=”top”} |
Ellipse |
|
and
terms have the same sign, different coefficients |
|
|
|
|
{: valign=”top”} |
Hyperbola |
|
and
terms have different signs, different coefficients |
|
|
{: valign=”top”}{: summary=”This table has three columns and five rows. The first row is a header row and it labels each column, “Conic,” “Characteristics of x squared and y squared terms,” and “Example.” The first column is a header column and it labels each row “Parabola,” “Circle,” “Ellipse,”, and “Hyperbola.” In row two, the Parabola is described as having either x squared or y squared and only one variable squared and the example is x is equal to 3 y squared minus 2 y plus 1. In row three, the Circle is described as having x squared and y squared terms with the same coefficients and the example is x squared plus y squared is equal to 49. In row four, the Ellipse is described as having x squared and y squared terms that have the same sign and different coefficients and the example is 4 x squared plus 25 y squared is equal to 100. In row five, the Hyperbola is described as having x squared and y squared terms that have different signs and different coefficients and the example is 25 y squared minus 4 x squared is equal to 100.” .unnumbered data-label=””}
Practice Makes Perfect
**Graph a Hyperbola with Center at
**
In the following exercises, graph.
![The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus two-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_312_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-fourths times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_314_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_316_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-fourths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_318_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-halves times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_320_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_322_img.jpg)
**Graph a Hyperbola with Center at
**
In the following exercises, graph.
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_324_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_326_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, negative 4) an asymptote that passes through (negative 7, 1) and (5, negative 9) and an asymptote that passes through (5, 1) and (negative 7, negative 9), and branches that pass through the vertices (1, 1) and (1, negative 9) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_328_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 1, 4) an asymptote that passes through (4, 8) and (negative 6, 0) and an asymptote that passes through (negative 6, 8) and (4, 0), and branches that pass through the vertices (negative 1, 0) and (negative 1, 8) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_330_img.jpg)
![The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (3, negative 2) an asymptote that passes through (8, 1) and (negative 2, negative 5) and an asymptote that passes through (negative 2, negative 1) and (8, negative 5), and branches that pass through the vertices (negative 2, negative 2) and (8, negative 2) and opens left and right.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_332_img.jpg)
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.
ⓐ
ⓑ
ⓒ
ⓓ
ⓐ
ⓑ
ⓒ
ⓓ
ⓐ parabola ⓑ circle ⓒ hyperbola ⓓ ellipse
Mixed Practice
In the following exercises, graph each equation.
![The graph shows the x y coordinate plane with a circle whose center is (2, negative 5) and whose radius is 6 units.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_340_img.jpg)
![The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 5) and co-vertices are (plus or minus 3, 0).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_342_img.jpg)
![The graph shows the x y coordinate plane with the center (1, 2) an asymptote that passes through (negative 2, 5) and (5, negative 1) and an asymptote that passes through (4, 5) and (2, 0), and branches that pass through the vertices (1, 5) and (negative 2, negative 1) and open up and down.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_344_img.jpg)
![The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 4) and co-vertices are (plus or minus 3, 0).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_11_04_346_img.jpg)
Writing Exercises
In your own words, define a hyperbola and write the equation of a hyperbola centered at the origin in standard form. Draw a sketch of the hyperbola labeling the center, vertices, and asymptotes.
Explain in your own words how to create and use the rectangle that helps graph a hyperbola.
Compare and contrast the graphs of the equations
and
Explain in your own words, how to distinguish the equation of an ellipse with the equation of a hyperbola.
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?
Glossary
- hyperbola
- A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.
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