### All Precalculus Resources

## Example Questions

### Example Question #44 : Graphs

Using the information below, determine the equation of the hyperbola.

Foci: and

Eccentricity:

**Possible Answers:**

**Correct answer:**

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci =

Distance between vertices =

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 12, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 4, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 12, we can conclude that

Center point:

Thus, the equation of the hyperbola is:

### Example Question #1 : Hyperbolas

Using the information below, determine the equation of the hyperbola.

Foci: and

Eccentricity:

**Possible Answers:**

**Correct answer:**

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci =

Distance between vertices =

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 8, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 8, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 8, we can conclude that

Center point:

Thus, the equation of the hyperbola is:

### Example Question #1 : Hyperbolas

What is the equation of the conic section graphed below

**Possible Answers:**

**Correct answer:**

The hyperbola pictured is centered at , meaning that the equation has a horizontal shift. The equation must have rather than just x. The hyperbola opens up and down, so the equation must be the y term minus the x term. The hyperbola is drawn according to the box going up/down 5 and left/right 2, so the y term must be or , and the x term must be or .

### Example Question #2 : Hyperbolas

How can this graph be changed to be the graph of

?

**Possible Answers:**

The graph should be an ellipse and not a hyperbola.

The graph should have -intercepts and not -intercepts.

The -intercepts should be at the points and .

The center box should extend up to and down to , stretching the graph.

The -intercepts should be at the points and .

**Correct answer:**

The -intercepts should be at the points and .

This equation should be thought of as .

This means that the hyperbola will be determined by a box with x-intercepts at and y-intercepts at .

The hyperbola was incorrectly drawn with the intercepts at instead.

### Example Question #3 : Hyperbolas

Which of the following would NOT be true of the graph for ?

**Possible Answers:**

The graph never intersects with the -axis.

The graph never intersects with the -axis.

The graph opens up and down.

All of these statements are true.

The graph is centered at .

**Correct answer:**

The graph never intersects with the -axis.

The graph should look like this:

### Example Question #4 : Hyperbolas

Which of these equations produce this graph, rotated 90 degrees?

**Possible Answers:**

**Correct answer:**

Rotated 90 degrees, this graph would be opening up and down instead of left and right, so the equation will have the y term minus the x term.

The box that the hyperbola is drawn around will also rotate. It will now be up/down 2 and left/right 3.

This makes the correct equation

.

### Example Question #46 : Graphs

Find the foci of the hyperbola with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with is first, that means the foci will lie on a horizontal transverse axis.

When the term with is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at and .

For a hyperbola with a vertical transverse access, the foci will be located at and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at and .

### Example Question #47 : Graphs

Find the foci of a hyperbola with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with is first, that means the foci will lie on a horizontal transverse axis.

When the term with is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at and .

For a hyperbola with a vertical transverse access, the foci will be located at and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at and .

### Example Question #5 : Hyperbolas

Find the foci of the hyperbola with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with is first, that means the foci will lie on a horizontal transverse axis.

When the term with is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at and .

For a hyperbola with a vertical transverse access, the foci will be located at and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at and .

### Example Question #48 : Graphs

Find the foci of the hyperbola with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with is first, that means the foci will lie on a horizontal transverse axis.

When the term with is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at and .

For a hyperbola with a vertical transverse access, the foci will be located at and .

Next, find .

The foci are then located at and .

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