MTH120 College Algebra Chapter 8.2

 8.2 The Hyperbola

A hyperbola is a type of conic section, similar to the ellipse and the parabola. It is defined as the set of all points in a plane such that the absolute difference of the distances to two fixed points, called the foci, is constant. This constant is represented by 2a, and it's called the major axis.

The standard equation for a hyperbola with its center at the origin (0,0) and the foci on the x-axis is given by:

$\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

Here, 'a' is the distance from the origin to the vertices on the transverse axis (along the x-axis), and 'b' is the distance from the origin to the vertices on the conjugate axis (along the y-axis). The values of 'a' and 'b' determine the shape and orientation of the hyperbola. If 'a' > 'b', the hyperbola opens horizontally, and if 'b' > 'a', it opens vertically.

The coordinates of the foci can be found using the values of 'a' and 'b' and are given by (+-c, 0), where 'c' is related to 'a' and 'b' by the equation:

$�=\sqrt{{�}^2+{�}^2}$

The vertices of the hyperbola are located at (+-a, 0) and (-+b, 0).

The asymptotes of the hyperbola are the lines that the hyperbola approaches as it extends infinitely in both directions. The equations of the asymptotes for a hyperbola with its center at the origin are:

$�=\pm \frac{�}{�}�$

To graph a hyperbola, you can use these key elements: the center, the vertices, the foci, and the asymptotes. The shape and orientation of the hyperbola can be determined from the equation and the values of 'a' and 'b'.

Hyperbolas can also have variations in their equations when the center is not at the origin, or when they are rotated, but the basic principles remain the same.

To locate the vertices and foci of a hyperbola, you'll need the equation of the hyperbola and some key information about its characteristics. The standard equation for a hyperbola with its center at the origin is:

$\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

Here, 'a' and 'b' are the parameters that determine the size and orientation of the hyperbola.

1. Finding the Vertices:

The vertices of a hyperbola are located along the transverse axis. For a hyperbola with its center at the origin, the vertices are at (+-a, 0). So, to locate the vertices, you simply find the values of 'a' and plot points (+a, 0) and (-a, 0) on the x-axis.

2. Finding the Foci:

The foci of a hyperbola are found using the relationship between 'a', 'b', and 'c', where 'c' is the distance from the center to each focus.

The formula for 'c' is:

$�=\sqrt{{�}^2+{�}^2}$

Once you have 'c,' you can locate the foci along the transverse axis, which is also the major axis of the hyperbola. For a hyperbola with its center at the origin, the foci are at (+-c, 0).

Here are the steps to locate the vertices and foci of a hyperbola:

1. Examine the equation to determine 'a' and 'b.'
2. Calculate 'c' using the formula $�=\sqrt{{�}^2+{�}^2}$.
3. Locate the vertices at (+a, 0) and (-a, 0) on the x-axis.
4. Locate the foci at (+c, 0) and (-c, 0) on the x-axis.

This will give you the positions of the vertices and foci of the hyperbola. Remember that if the center of the hyperbola is not at the origin or if it's rotated, the process will be slightly different, and you'll need to consider the center's coordinates and the orientation of the axes.

Deriving the equation of a hyperbola centered at the origin is relatively straightforward. The standard equation for a hyperbola centered at the origin and aligned with the coordinate axes is:

$\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

Here's how you can derive this equation:

1. Consider the Definition of a Hyperbola:

   A hyperbola is defined as the set of all points such that the absolute difference of their distances to two fixed points (the foci) is constant. The line connecting the foci is called the transverse axis, and the midpoint of this line is the center of the hyperbola.

2. Position of the Foci:

   For a hyperbola centered at the origin, the foci lie along the x-axis, and their coordinates are (+-c, 0). You need to find 'c', which is the distance from the origin to each focus.

3. Use the Distance Formula:

   The distance between a point (x, y) and the focus (+c, 0) is given by the distance formula:

   $\sqrt{(�-�{)}^2+{�}^2}$

   The distance between the same point and the other focus (-c, 0) is also:

   $\sqrt{(�+�{)}^2+{�}^2}$

4. Set Up the Equation:

   According to the definition of a hyperbola, these two distances must have a constant difference, which is 2a:

   $\sqrt{(�-�{)}^2+{�}^2}-\sqrt{(�+�{)}^2+{�}^2}=2�$

5. Simplify the Equation:

   To get the standard equation, we need to isolate the radical terms on one side:

   $\sqrt{(�-�{)}^2+{�}^2}=2�+\sqrt{(�+�{)}^2+{�}^2}$

   Now, square both sides to eliminate the square roots:

   $(�-�{)}^2+{�}^2=(2�+\sqrt{(�+�{)}^2+{�}^2}{)}^2$

6. Simplify Further:

   Expand and simplify both sides:

   ${�}^2-2��+{�}^2+{�}^2=4{�}^2+4�\sqrt{(�+�{)}^2+{�}^2}+(�+�{)}^2+{�}^2$

   Cancel out the ${�}^2$ terms:

   ${�}^2-2��+{�}^2=4{�}^2+4�\sqrt{(�+�{)}^2+{�}^2}+(�+�{)}^2$

7. Isolate the Radical Term:

   Move the terms without radicals to one side:

   ${�}^2-2��+{�}^2-(�+�{)}^2=4{�}^2+4�\sqrt{(�+�{)}^2+{�}^2}$

   Simplify the left side:

   ${�}^2-2��+{�}^2-({�}^2+2��+{�}^2)=4{�}^2+4�\sqrt{(�+�{)}^2+{�}^2}$

   This simplifies to:

   $-4��=4{�}^2+4�\sqrt{(�+�{)}^2+{�}^2}$

8. Divide by 4a:

   $�(-�\mathrm{/}�)=1+\sqrt{(�+�{)}^2+{�}^2}$

9. Square Both Sides Again:

   ${�}^2\left({�}^2\mathrm{/}{�}^2\right)=(1+\sqrt{(�+�{)}^2+{�}^2}{)}^2$

   Expanding and simplifying the right side:

   ${�}^2\left({�}^2\mathrm{/}{�}^2\right)=1+2\sqrt{(�+�{)}^2+{�}^2}+(�+�{)}^2+{�}^2$

10. Simplify Further:

    ${�}^2\left({�}^2\mathrm{/}{�}^2\right)=1+2\sqrt{(�+�{)}^2+{�}^2}+{�}^2+2��+{�}^2+{�}^2$

    Cancel out ${�}^2$ terms and subtract $1$ from both sides:

    $\left({�}^2\mathrm{/}{�}^2\right)-1=2\sqrt{(�+�{)}^2+{�}^2}+2��$

11. Isolate the Radical Term Again:

    $\left({�}^2\mathrm{/}{�}^2\right)-1-2��=2\sqrt{(�+�{)}^2+{�}^2}$

12. Square Both Sides Once More:

    ${(\left({�}^2\mathrm{/}{�}^2\right)-1-2��)}^2=4((�+�{)}^2+{�}^2)$

13. Expand and Simplify:

    ${\left({�}^2\mathrm{/}{�}^2\right)}^2-2\left({�}^2\mathrm{/}{�}^2\right)-2�\left({�}^2\mathrm{/}{�}^2\right)�+1+4��+4{�}^2+4{�}^2=4{�}^2+8��+4{�}^2+4{�}^2$

    Cancel out common terms and simplify:

    $\left({�}^4\mathrm{/}{�}^4\right)-2\left({�}^2\mathrm{/}{�}^2\right)-2�\left({�}^2\mathrm{/}{�}^2\right)�+1=4{�}^2$

14. Rearrange Terms:

    $\left({�}^4\mathrm{/}{�}^4\right)-2\left({�}^2\mathrm{/}{�}^2\right)-4{�}^2=2�\left({�}^2\mathrm{/}{�}^2\right)�-1$

15. Divide by $2�\left({�}^2\mathrm{/}{�}^2\right)$:

    $\frac{\left({�}^4\mathrm{/}{�}^4\right)-2\left({�}^2\mathrm{/}{�}^2\right)-4{�}^2}{2�\left({�}^2\mathrm{/}{�}^2\right)}=\frac{2�\left({�}^2\mathrm{/}{�}^2\right)�-1}{2�\left({�}^2\mathrm{/}{�}^2\right)}$

16. Simplify:

    $\frac{{�}^4-2{�}^2{�}^2-4{�}^2{�}^2}{2{�}^3}=\frac{�}{{�}^2}-\frac{1}{2{�}^3}$

The standard form of the equation of a hyperbola with its center at the origin (0,0) can be written in two ways, depending on the orientation of the hyperbola:

1. Horizontal Hyperbola: The equation for a horizontal hyperbola with center (0,0) is:

   $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

   • 'a' is the distance from the center to the vertices along the x-axis (transverse axis).
   • 'b' is the distance from the center to the vertices along the y-axis (conjugate axis).
   • The foci lie at (+-c, 0), where 'c' is calculated as $�=\sqrt{{�}^2+{�}^2}$.
   • The asymptotes are the lines $�=\pm \frac{�}{�}�$.

2. Vertical Hyperbola: The equation for a vertical hyperbola with center (0,0) is:

   $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

   • 'a' is the distance from the center to the vertices along the y-axis (transverse axis).
   • 'b' is the distance from the center to the vertices along the x-axis (conjugate axis).
   • The foci lie at (0, +-c), where 'c' is calculated as $�=\sqrt{{�}^2+{�}^2}$.
   • The asymptotes are the lines $�=\pm \frac{�}{�}�$.

In both forms, 'a' and 'b' determine the size and shape of the hyperbola, while 'c' is related to them through the Pythagorean relationship ${�}^2={�}^2+{�}^2$. The asymptotes are the lines that the hyperbola approaches as it extends infinitely in both directions.

These standard forms are very useful for graphing and understanding the properties of hyperbolas when the center is at the origin. If the center is located at a point other than (0,0), the equation of the hyperbola will be slightly different, involving translations and rotations.

To locate the vertices and foci of a hyperbola given its equation in standard form, you can use the following steps:

1. Identify 'a' and 'b' from the equation:

   For the standard form of a hyperbola with its center at the origin, the equation is:

   $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

   From this equation:

   • 'a' is the square root of the denominator under the x-term.
   • 'b' is the square root of the denominator under the y-term.

2. Calculate 'c':

   'c' represents the distance from the center to the foci and is related to 'a' and 'b' by the equation:

   $�=\sqrt{{�}^2+{�}^2}$

3. Locate the vertices:

   The vertices of the hyperbola are located along the transverse axis, which is the axis aligned with the major axis. For a hyperbola centered at the origin, the vertices are at (+-a, 0).

4. Locate the foci:

   The foci are also located along the major axis. For a hyperbola centered at the origin, the foci are at (+-c, 0).

So, to summarize, the vertices of the hyperbola are at (+-a, 0), and the foci are at (+-c, 0). These steps help you locate the essential points of a hyperbola based on its standard form equation.

To write the equation of a hyperbola in standard form, you need to have specific information about the hyperbola, particularly the location of its center, the lengths of its major and minor axes, and whether it's horizontal or vertical. The general forms for the standard equations of hyperbolas are:

1. Horizontal Hyperbola with Center at (h, k):

   $\frac{(�-ℎ{)}^2}{{�}^2}-\frac{(�-�{)}^2}{{�}^2}=1$

   • (h, k) represents the coordinates of the center.
   • 'a' is the distance from the center to the vertices along the x-axis (transverse axis).
   • 'b' is the distance from the center to the vertices along the y-axis (conjugate axis).

2. Vertical Hyperbola with Center at (h, k):

   $\frac{(�-�{)}^2}{{�}^2}-\frac{(�-ℎ{)}^2}{{�}^2}=1$

   • (h, k) represents the coordinates of the center.
   • 'a' is the distance from the center to the vertices along the y-axis (transverse axis).
   • 'b' is the distance from the center to the vertices along the x-axis (conjugate axis).

To write the equation of a specific hyperbola in standard form, follow these steps:

1. Determine whether the hyperbola is horizontal or vertical based on its orientation.

2. Identify the center of the hyperbola. This is usually given or can be found in the problem.

3. Find 'a' and 'b' based on the problem statement, which often involves information about the vertices, foci, or other relevant details.

4. Plug in the values of 'h', 'k', 'a', and 'b' into the appropriate standard form equation (horizontal or vertical) to obtain the equation for the hyperbola.

5. Simplify and rearrange the equation if necessary.

Here's an example:

Example: Write the equation of the hyperbola with a center at (2, -3), horizontal orientation, a major axis of length 6, and a minor axis of length 4.

1. The hyperbola is horizontal.
2. The center is at (h, k) = (2, -3).
3. The length of the major axis is 6, so 'a' is 3 (half of 6), and the length of the minor axis is 4, so 'b' is 2 (half of 4).

Now, we can use the standard form equation for a horizontal hyperbola:

$\frac{(�-2{)}^2}{3^2}-\frac{(�+3{)}^2}{2^2}=1$

Simplifying further:

$\frac{(�-2{)}^2}{9}-\frac{(�+3{)}^2}{4}=1$

So, the equation of the hyperbola is in standard form.

A hyperbola centered at the origin (0,0) is one that is symmetric with respect to the x-axis and y-axis. The standard equations for hyperbolas with center at the origin are:

1. Horizontal Hyperbola: The equation for a horizontal hyperbola centered at the origin is:

   $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

   • 'a' is the distance from the origin to the vertices along the x-axis (transverse axis).
   • 'b' is the distance from the origin to the vertices along the y-axis (conjugate axis).

   The foci lie at (+-c, 0), where 'c' is calculated as $�=\sqrt{{�}^2+{�}^2}$.

   The asymptotes are the lines $�=\pm \frac{�}{�}�$.

2. Vertical Hyperbola: The equation for a vertical hyperbola centered at the origin is:

   $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

   • 'a' is the distance from the origin to the vertices along the y-axis (transverse axis).
   • 'b' is the distance from the origin to the vertices along the x-axis (conjugate axis).

   The foci lie at (0, +-c), where 'c' is calculated as $�=\sqrt{{�}^2+{�}^2}$.

   The asymptotes are the lines $�=\pm \frac{�}{�}�$.

In both forms, 'a' and 'b' determine the size and shape of the hyperbola, while 'c' is related to them through the Pythagorean relationship ${�}^2={�}^2+{�}^2$.

These standard forms are very useful for graphing and understanding the properties of hyperbolas when the center is at the origin. If the center is located at a point other than (0,0), the equation of the hyperbola will be slightly different, involving translations and rotations.

Hyperbolas that are not centered at the origin have equations that are slightly more complex than those centered at the origin. To find the equation of a hyperbola that is not centered at the origin, you'll need to consider the location of the center, the lengths of the major and minor axes, and the orientation of the hyperbola. Here's how to write the equation for such hyperbolas:

1. Determine the Coordinates of the Center (h, k): The center of the hyperbola is represented by the coordinates (h, k).

2. Determine 'a' and 'b':

   • 'a' is the distance from the center to a vertex along the major axis.
   • 'b' is the distance from the center to a vertex along the minor axis.
   • These values are usually provided in the problem or can be calculated from the given information.

3. Identify the Orientation: Determine whether the hyperbola is horizontal or vertical based on the given information.

4. Use the Appropriate Standard Form:

   • Horizontal Hyperbola: If the hyperbola is horizontal, the standard equation is: $\frac{(�-ℎ{)}^2}{{�}^2}-\frac{(�-�{)}^2}{{�}^2}=1$

   • Vertical Hyperbola: If the hyperbola is vertical, the standard equation is: $\frac{(�-�{)}^2}{{�}^2}-\frac{(�-ℎ{)}^2}{{�}^2}=1$

5. Adjust the Center Terms: If the center of the hyperbola is not at the origin (0,0), be sure to include the values of 'h' and 'k' in the equation.

   • Replace 'h' with the x-coordinate of the center.
   • Replace 'k' with the y-coordinate of the center.

6. Simplify the Equation: If needed, simplify the equation further by multiplying or dividing through by a constant to put it into its most concise form.

Here's an example:

Example: Write the equation for a hyperbola with a center at (3, -2), a horizontal major axis of length 8, and a vertical minor axis of length 6.

1. The center is (h, k) = (3, -2).
2. The length of the major axis is 8, so 'a' is 4 (half of 8), and the length of the minor axis is 6, so 'b' is 3 (half of 6).
3. The hyperbola is both horizontal and vertical.
4. For a horizontal hyperbola: $\frac{(�-3{)}^2}{4^2}-\frac{(�+2{)}^2}{3^2}=1$

This is the equation for the hyperbola with a center at (3, -2), a horizontal major axis of length 8, and a vertical minor axis of length 6.

The standard forms of the equation of a hyperbola with its center at (h, k) involve both horizontal and vertical variations, and they depend on the orientation of the hyperbola. Here are the standard forms and examples for each case:

1. Horizontal Hyperbola:

   The standard equation for a horizontal hyperbola with its center at (h, k) is:

   $\frac{(�-ℎ{)}^2}{{�}^2}-\frac{(�-�{)}^2}{{�}^2}=1$

   • 'a' is the distance from the center to the vertices along the x-axis (transverse axis).
   • 'b' is the distance from the center to the vertices along the y-axis (conjugate axis).

Example 1: Write the equation of a horizontal hyperbola with its center at (2, -1), a = 3, and b = 2.

The standard equation is:

$\frac{(�-2{)}^2}{3^2}-\frac{(�+1{)}^2}{2^2}=1$

2. Vertical Hyperbola:

   The standard equation for a vertical hyperbola with its center at (h, k) is:

   $\frac{(�-�{)}^2}{{�}^2}-\frac{(�-ℎ{)}^2}{{�}^2}=1$

   • 'a' is the distance from the center to the vertices along the y-axis (transverse axis).
   • 'b' is the distance from the center to the vertices along the x-axis (conjugate axis).

Example 2: Write the equation of a vertical hyperbola with its center at (1, -2), a = 4, and b = 3.

The standard equation is:

$\frac{(�+2{)}^2}{4^2}-\frac{(�-1{)}^2}{3^2}=1$

These standard forms are useful for graphing hyperbolas and understanding their properties when the center is not at the origin. The values of 'h', 'k', 'a', and 'b' are used to determine the size, orientation, and position of the hyperbola.

To write the equation of a hyperbola in standard form when given the vertices and foci and the center coordinates (h, k), you can follow these steps:

Let's assume you have the coordinates of the vertices, which are ($ℎ\pm �,�$), and the coordinates of the foci, which are ($ℎ\pm �,�$).

1. Determine 'a' and 'c':

   • The value of 'a' is the distance from the center to a vertex along the major axis: $�=\mathrm{\mid }�\mathrm{\mid }=\mathrm{\mid }ℎ\pm �-ℎ\mathrm{\mid }$.
   • The value of 'c' is the distance from the center to a focus along the major axis: $�=\mathrm{\mid }�\mathrm{\mid }=\mathrm{\mid }ℎ\pm �-ℎ\mathrm{\mid }$.

2. Determine 'b':

   • The value of 'b' can be found using the relationship ${�}^2={�}^2+{�}^2$.
   • Solve for 'b': $�=\sqrt{{�}^2-{�}^2}$.

3. Determine the orientation:

   • Observe whether the major axis is horizontal or vertical.

4. Use the appropriate standard form:

   • Horizontal Major Axis (vertices along the x-axis):

     • The standard form for a horizontal hyperbola with center at (h, k) is: $\frac{(�-ℎ{)}^2}{{�}^2}-\frac{(�-�{)}^2}{{�}^2}=1$

   • Vertical Major Axis (vertices along the y-axis):

     • The standard form for a vertical hyperbola with center at (h, k) is: $\frac{(�-�{)}^2}{{�}^2}-\frac{(�-ℎ{)}^2}{{�}^2}=1$

5. Substitute the values of 'a', 'b', 'h', and 'k' into the appropriate standard form equation to write the standard form equation of the hyperbola.

Here's an example:

Example: Write the equation for a hyperbola with its center at (2, 3), vertices at (3, 3) and (7, 3), and foci at (4, 3) and (6, 3).

1. Calculate 'a' and 'c':

   • $�=\mathrm{\mid }3-2\mathrm{\mid }=1$
   • $�=\mathrm{\mid }4-2\mathrm{\mid }=2$

2. Calculate 'b':

   • Use the relationship ${�}^2={�}^2+{�}^2$:
   • $4=1+{�}^2$
   • $�=\sqrt{3}$

3. Determine the orientation: The major axis is horizontal.

4. Use the standard form for a horizontal hyperbola: $\frac{(�-2{)}^2}{1^2}-\frac{(�-3{)}^2}{(\sqrt{3}{)}^2}=1$

This is the equation for the hyperbola with the given center, vertices, and foci.

To graph hyperbolas centered at the origin, you can follow these steps. I'll provide two examples to illustrate how to graph horizontal and vertical hyperbolas.

Example 1: Graphing a Horizontal Hyperbola

Suppose you want to graph the hyperbola with the equation:

$\frac{{�}^2}{9}-\frac{{�}^2}{4}=1$

1. Determine 'a' and 'b':

   • 'a' is the square root of the denominator under the x-term: $�=\sqrt{9}=3$.
   • 'b' is the square root of the denominator under the y-term: $�=\sqrt{4}=2$.

2. Identify the orientation: This is a horizontal hyperbola because the term with 'x' is positive.

3. Find the vertices and foci:

   • Vertices are at (+-a, 0): Vertices = (+-3, 0).
   • Foci can be calculated using $�=\sqrt{{�}^2+{�}^2}$: $�=\sqrt{9+4}=\sqrt{13}$.
   • Foci are at (+-c, 0): Foci = (+-√13, 0).

4. Determine the asymptotes:

   • The asymptotes for horizontal hyperbolas have the equations $�=\pm \frac{�}{�}�$: Asymptotes = $�=\pm \frac{2}{3}�$.

5. Plot the vertices, foci, and asymptotes on your graph.

6. Sketch the hyperbola using the vertices and asymptotes as guides. The hyperbola should curve away from the center (origin) and approach the asymptotes as it extends.

Example 2: Graphing a Vertical Hyperbola

Suppose you want to graph the hyperbola with the equation:

$\frac{{�}^2}{9}-\frac{{�}^2}{4}=1$

1. Determine 'a' and 'b':

   • 'a' is the square root of the denominator under the y-term: $�=\sqrt{9}=3$.
   • 'b' is the square root of the denominator under the x-term: $�=\sqrt{4}=2$.

2. Identify the orientation: This is a vertical hyperbola because the term with 'y' is positive.

3. Find the vertices and foci:

   • Vertices are at (0, +-a): Vertices = (0, +-3).
   • Foci can be calculated using $�=\sqrt{{�}^2+{�}^2}$: $�=\sqrt{9+4}=\sqrt{13}$.
   • Foci are at (0, +-c): Foci = (0, +-√13).

4. Determine the asymptotes:

   • The asymptotes for vertical hyperbolas have the equations $�=\pm \frac{�}{�}�$: Asymptotes = $�=\pm \frac{3}{2}�$.

5. Plot the vertices, foci, and asymptotes on your graph.

6. Sketch the hyperbola using the vertices and asymptotes as guides. The hyperbola should curve away from the center (origin) and approach the asymptotes as it extends.

Remember to label your axes and the key points (vertices and foci) on your graph to create an accurate representation of the hyperbola.

Graphing hyperbolas that are not centered at the origin involves a few additional steps compared to hyperbolas centered at the origin. To graph these hyperbolas, you'll need to consider the location of the center (h, k), the lengths of the major and minor axes, and the orientation (horizontal or vertical). Here are the steps with two examples: one for a hyperbola with a horizontal major axis and one for a hyperbola with a vertical major axis.

Example 1: Graphing a Hyperbola with Horizontal Major Axis

Suppose you want to graph the hyperbola with the equation:

$\frac{(�-2{)}^2}{16}-\frac{(�+1{)}^2}{9}=1$

1. Identify 'h' and 'k':

   • 'h' is 2 (the x-coordinate of the center).
   • 'k' is -1 (the y-coordinate of the center).

2. Determine 'a' and 'b':

   • 'a' is the square root of the denominator under the x-term: $�=\sqrt{16}=4$.
   • 'b' is the square root of the denominator under the y-term: $�=\sqrt{9}=3$.

3. Identify the orientation: This is a hyperbola with a horizontal major axis because the term with 'x' is positive.

4. Find the vertices and foci:

   • Vertices are at (h ± a, k): Vertices = (2 ± 4, -1) = (-2, -1) and (6, -1).
   • Foci can be calculated using $�=\sqrt{{�}^2+{�}^2}$: $�=\sqrt{16+9}=\sqrt{25}=5$.
   • Foci are at (h ± c, k): Foci = (2 ± 5, -1) = (-3, -1) and (7, -1).

5. Determine the asymptotes:

   • The asymptotes for horizontal hyperbolas have the equations $�=�\pm \frac{�}{�}(�-ℎ)$: Asymptotes = $�=-1\pm \frac{3}{4}(�-2)$.

6. Plot the vertices, foci, and asymptotes on your graph.

7. Sketch the hyperbola using the vertices and asymptotes as guides. The hyperbola should curve away from the center and approach the asymptotes as it extends.

Example 2: Graphing a Hyperbola with Vertical Major Axis

Suppose you want to graph the hyperbola with the equation:

$\frac{(�-1{)}^2}{9}-\frac{(�+2{)}^2}{4}=1$

1. Identify 'h' and 'k':

   • 'h' is -2 (the x-coordinate of the center).
   • 'k' is 1 (the y-coordinate of the center).

2. Determine 'a' and 'b':

   • 'a' is the square root of the denominator under the y-term: $�=\sqrt{9}=3$.
   • 'b' is the square root of the denominator under the x-term: $�=\sqrt{4}=2$.

3. Identify the orientation: This is a hyperbola with a vertical major axis because the term with 'y' is positive.

4. Find the vertices and foci:

   • Vertices are at (h, k ± a): Vertices = (-2, 1 ± 3) = (-2, 4) and (-2, -2).
   • Foci can be calculated using $�=\sqrt{{�}^2+{�}^2}$: $�=\sqrt{9+4}=\sqrt{13}$.
   • Foci are at (h, k ± c): Foci = (-2, 1 ± √13) ≈ (-2, 4.6) and (-2, -2.6).

5. Determine the asymptotes:

   • The asymptotes for vertical hyperbolas have the equations $�=�\pm \frac{�}{�}(�-ℎ)$: Asymptotes = $�=1\pm \frac{3}{2}(�+2)$.

6. Plot the vertices, foci, and asymptotes on your graph.

7. Sketch the hyperbola using the vertices and asymptotes as guides. The hyperbola should curve away from the center and approach the asymptotes as it extends.

Remember to label your axes and the key points (vertices and foci) on your graph to create an accurate representation of the hyperbola.

Solving applied problems involving hyperbolas typically requires you to understand the properties of hyperbolas and use their equations to model real-world situations. Here are two examples of applied problems involving hyperbolas:

Example 1: Satellite Communication

Suppose you are designing a satellite communication system, and you want to determine the range at which the satellite can communicate with a ground station. You know that the satellite is in a geostationary orbit, which forms a circular path around the Earth with a radius of 42,164 kilometers. The satellite can communicate with the ground station when it is within a certain range of angles from the zenith (directly overhead).

The problem involves a hyperbola because the satellite can communicate with the ground station within a specific range of angles. You can use the properties of a hyperbola to solve this problem.

Solution:

1. Identify the center and key parameters:

   • The satellite is in a circular geostationary orbit, which forms the center of the hyperbola.
   • The radius of the circular orbit is 42,164 kilometers, which is the distance from the center to either vertex of the hyperbola (a).

2. Write the equation of the hyperbola: The standard form equation for a hyperbola centered at the origin is $\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$. In this case, 'a' is the radius of the circular orbit, so the equation becomes $\frac{{�}^2}{42164^2}-\frac{{�}^2}{{�}^2}=1$.

3. Determine the range of angles:

   • The range of angles from the zenith within which the satellite can communicate corresponds to the hyperbola. The values of 'a' and 'b' are known from the circular orbit.
   • Calculate 'b' using the relationship ${�}^2={�}^2+{�}^2$, where 'c' is the distance from the center to a focus.
   • Substitute 'a' and 'c' into the equation to determine 'b'.

4. Solve for 'y' when $�=�$ and $�=-�$ to find the range of angles from the zenith.

5. Interpret the results: You will have the range of angles within which the satellite can communicate with the ground station.

Example 2: Fireworks Display

Suppose you are organizing a fireworks display, and you want to launch fireworks from a point (2, 3) on the ground to create a spectacle in the night sky. The fireworks travel in a hyperbolic path, and you want to determine the equation of the hyperbola to ensure the fireworks explode at the desired height and distance.

Solution:

1. Identify the key parameters:

   • The point (2, 3) represents the center of the hyperbola.
   • You need to determine 'a' and 'b' to write the equation.

2. Write the equation of the hyperbola: The standard form equation for a hyperbola with the center at (h, k) is $\frac{(�-ℎ{)}^2}{{�}^2}-\frac{(�-�{)}^2}{{�}^2}=1$.

3. Determine 'a' and 'b':

   • The desired explosion point (2, 3) is on the hyperbola, which represents a vertex.
   • 'a' is the distance from the center to the vertex, so calculate 'a' using the distance formula from the center to the point (2, 3).

4. Determine 'b':

   • 'b' is related to 'a' and the distance from the center to a focus using the relationship ${�}^2={�}^2+{�}^2$.
   • Calculate 'c' as the distance from the center to a focus, which is usually provided in the problem.

5. Write the equation of the hyperbola with the known values of 'a' and 'b'.

6. Determine the trajectory of the fireworks: You will have the equation that models the path of the fireworks in the night sky.

These examples illustrate how to apply the properties of hyperbolas to real-world problems by using the standard form equations and understanding the relationships between key parameters.