Hyperbolic Functions - Algebra 2
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What are the vertices of this hyperbola?
What are the vertices of this hyperbola?
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The first thing we need to find for this hyperbola is the center. This is simply the point where 
and 
both equal 
, which is 
. Since the 
term is the positive one, the hyperbola opens horizontally, which means we need to look at the denominator of that 
term.
The denominator is 
which is 
, so our vertices are 
, or
and 
.
The first thing we need to find for this hyperbola is the center. This is simply the point where
The denominator is
Which of the following answers best represent 
?
Which of the following answers best represent
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The correct definition of hyperbolic sine is:
Therefore, by multiplying 2 by both sides we get the following answer,
The correct definition of hyperbolic sine is:
Therefore, by multiplying 2 by both sides we get the following answer,
In which direction does the graph of the above hyperbola open?
In which direction does the graph of the above hyperbola open?
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To determine which direction a hyperbola opens, first get the equation into standard form for a conic section:
This equation gives us a hyperbola when the coefficient in front of either the x-squared or the y-squared term (but not both!) is negative. In this problem, the coefficient in front of the x-squared term is positive, but the coefficient of the y-squared term is negative. Here are the rules for hyperbola directions:
- If the coefficient of the x-squared term is positive but the coefficient of the y-squared term is negative, this is a hyperbola that opens horizontally.
- If the coefficient of the x-squared term is negative but the coefficient of the y-squared term is positive, this is a hyperbola that opens vertically.
Thus, we have a graph of a horizontal hyperbola.
To determine which direction a hyperbola opens, first get the equation into standard form for a conic section:
This equation gives us a hyperbola when the coefficient in front of either the x-squared or the y-squared term (but not both!) is negative. In this problem, the coefficient in front of the x-squared term is positive, but the coefficient of the y-squared term is negative. Here are the rules for hyperbola directions:
- If the coefficient of the x-squared term is positive but the coefficient of the y-squared term is negative, this is a hyperbola that opens horizontally.
- If the coefficient of the x-squared term is negative but the coefficient of the y-squared term is positive, this is a hyperbola that opens vertically.
Thus, we have a graph of a horizontal hyperbola.
Write the expression for this hyperbola in standard form:
Write the expression for this hyperbola in standard form:
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The standard form of a hyperbola is
or the similar form with a positive 
term and negative 
term. So to start out getting this equation in standard form, we must complete the square on the quadratics in 
and 
.
the coefficient of 
is 
, so completing the square we get
and similarly with 
we get
and so our starting expression can be written as
Dividing by 
on both sides we get the standard representation of the hyperbola,
The standard form of a hyperbola is
or the similar form with a positive
the coefficient of
and similarly with
and so our starting expression can be written as
Dividing by
Find the vertices of the following hyperbolic function:
Find the vertices of the following hyperbolic function:
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We start by noticing that our hyperbola is given in the following form:
In order to determine the vertices of the hyperbola, we must first locate its center. Using the standard form given above, we know the center of the hyperbola occurs at the point (h,k), so for the equation given in the problem the center is at (2,-3). Now that we know the location of the hyperbola's center, our next step is to determine how far the vertices are from the center of the hyperbola. Looking at our equation, we can see it is in the form where the x term occurs first, which means the hyperbola opens left and right as opposed to up and down (which would be the case if the y term occurred first. Given this information, we know the vertices of the hyperbola are going to be a distance 
to the left and right of the center. The denominator 
of the x term in the hyperbolic equation is 16, which means 
is equal to 4, so the vertices of the hyperbola will be 4 units to the left and right of the center (2,-3), which gives us:
and 
We start by noticing that our hyperbola is given in the following form:
In order to determine the vertices of the hyperbola, we must first locate its center. Using the standard form given above, we know the center of the hyperbola occurs at the point (h,k), so for the equation given in the problem the center is at (2,-3). Now that we know the location of the hyperbola's center, our next step is to determine how far the vertices are from the center of the hyperbola. Looking at our equation, we can see it is in the form where the x term occurs first, which means the hyperbola opens left and right as opposed to up and down (which would be the case if the y term occurred first. Given this information, we know the vertices of the hyperbola are going to be a distance
Which of the following equations represents a vertical hyperbola with a center of 
and asymptotes at 
?
Which of the following equations represents a vertical hyperbola with a center of
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First, we need to become familiar with the standard form of a hyperbolic equation:
The center is always at 
. This means that for this problem, the numerators of each term will have to contain 
and 
.
To determine if a hyperbola opens vertically or horizontally, look at the sign of each variable. A vertical parabola has a positive 
term; a horizontal parabola has a positive 
term. In this case, we need a vertical parabola, so the 
term will have to be positive.
(NOTE: If both terms are the same sign, you have an ellipse, not a parabola.)
The asymptotes of a parabola are always found by the equation 
, where 
is found in the denominator of the 
term and 
is found in the denominator of the 
term. Since our asymptotes are 
, we know that 
must be 4 and 
must be 3. That means that the number underneath the 
term has to be 16, and the number underneath the 
term has to be 9.
First, we need to become familiar with the standard form of a hyperbolic equation:
The center is always at
To determine if a hyperbola opens vertically or horizontally, look at the sign of each variable. A vertical parabola has a positive
(NOTE: If both terms are the same sign, you have an ellipse, not a parabola.)
The asymptotes of a parabola are always found by the equation
What is the shape of the graph depicted by the equation:
What is the shape of the graph depicted by the equation:
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The standard equation of a hyperbola is:
The standard equation of a hyperbola is:
Express the following hyperbolic function in standard form:
Express the following hyperbolic function in standard form:
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In order to express the given hyperbolic function in standard form, we must write it in one of the following two ways:
From our formulas for the standard form of a hyperbolic equation above, we can see that the term on the right side of the equation is always 1, so we must divide both sides of the given equation by 52, which gives us:
Simplifying, we obtain our final answer in standard form:
In order to express the given hyperbolic function in standard form, we must write it in one of the following two ways:
From our formulas for the standard form of a hyperbolic equation above, we can see that the term on the right side of the equation is always 1, so we must divide both sides of the given equation by 52, which gives us:
Simplifying, we obtain our final answer in standard form:
Which of the following best represents 
, if the value of 
is zero?
Which of the following best represents
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Find the values of hyperbolic sine and cosine when x is zero. According to the properties:
Therefore:
Find the values of hyperbolic sine and cosine when x is zero. According to the properties:
Therefore:
What is the value of 
?
What is the value of
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The hyperbolic tangent will need to be rewritten in terms of hyperbolic sine and cosine.
According to the properties:
Therefore:
The hyperbolic tangent will need to be rewritten in terms of hyperbolic sine and cosine.
According to the properties:
Therefore:
Simplify: 
Simplify:
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The following is a property of hyperbolics that is closely similar to the problem.
We will need to rewrite this equation by taking a negative one as the common factor, and divide the negative one on both sides.
Substitute the value into the problem.
The following is a property of hyperbolics that is closely similar to the problem.
We will need to rewrite this equation by taking a negative one as the common factor, and divide the negative one on both sides.
Substitute the value into the problem.
Which of the following is the correct expression for a hyperbola that is shifted 
units up and 
to the right of 
?
Which of the following is the correct expression for a hyperbola that is shifted
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The parent function of a hyperbola is represented by the function 
where 
is the center of the hyperbola. To shift the original function up by 
simply add 
. To shift it to the right 
take away 
.
The parent function of a hyperbola is represented by the function
Find the foci of the hyperbola: 
Find the foci of the hyperbola:
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Write the standard forms for a hyperbola.
OR:
The standard form is given in the second case, which will have different parameters compared to the first form.
Center: 
Foci: 
, where 
Identify the coefficients 
and substitute to find the value of 
.
The answer is: 
Write the standard forms for a hyperbola.
OR:
The standard form is given in the second case, which will have different parameters compared to the first form.
Center:
Foci:
Identify the coefficients
The answer is:
Given the hyperbola 
, what is the 
value of the center?
Given the hyperbola
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In order to determine the center, we will first need to rewrite this equation in standard form.
Isolate 41 on the right side. Subtract 
and add 
on both sides.
The equation becomes:
Group the x and y terms. Be careful of the negative signs.
Pull out a common factor of 4 on the second parentheses.
Complete the square twice. Divide the second term of each parentheses by two and square the quantity. Add the terms on both sides.
This equation becomes:
Factorize the left side and simplify the right.
Divide both sides by nine.
The equation is now in the standard form of a hyperbola.
The center is at: 
The answer is: 
In order to determine the center, we will first need to rewrite this equation in standard form.
Isolate 41 on the right side. Subtract
The equation becomes:
Group the x and y terms. Be careful of the negative signs.
Pull out a common factor of 4 on the second parentheses.
Complete the square twice. Divide the second term of each parentheses by two and square the quantity. Add the terms on both sides.
This equation becomes:
Factorize the left side and simplify the right.
Divide both sides by nine.
The equation is now in the standard form of a hyperbola.
The center is at:
The answer is:
In which direction does the graph of the hyperbola 
open?
In which direction does the graph of the hyperbola 
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The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. The x-term appears first, so the given equation represents a horizontal hyperbola.
The equation for a horizontal hyperbola is 
Which of the following shapes does the graph of the equation 
take?
Which of the following shapes does the graph of the equation 
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The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. Both include x- and y-terms combined using subtraction. The equation for an ellipse is 
. Because the given equation connects the x- and y-terms using addition rather than subtraction, it represents an ellipse rather than a hyperbola. If the equation took the form 
, it would represent a circle. If the equation took the form 
, it would represent a parabola.
The equation for a horizontal hyperbola is 
Which of the following shapes does the graph of the equation 
take?
Which of the following shapes does the graph of the equation 
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The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. Both include x- and y-terms combined using subtraction. If the equation took the form 
(using addition rather than subtraction to combine the x- and y-terms), it would represent an ellipse. If the equation took the form 
, it would represent a circle. If the equation took the form 
, it would represent a parabola.
The equation for a horizontal hyperbola is 
What are the coordinates of the center of the hyperbolic inequality 
?
What are the coordinates of the center of the hyperbolic inequality 
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The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. In both, (h, v) is the center of the hyperbola. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. Because the two terms are combined using subtraction and the y-term appears first, this inequality represents a vertical hyperbola. To derive the center of a hyperbola from its equation or inequality, flip the sign of the constants that appear after the x and y in the equation or inequality. The constant following x is -1, so the x-coordinate of the center is 1. The constant following y is 2, so the y-coordinate of the center is -2.
The graph of the hyperbolic inequality 
appears as follows:
With vertices of (1, 1) and (1, -5), you can see that the midpoint between them is (1, -2).
The equation for a horizontal hyperbola is 
The graph of the hyperbolic inequality 
With vertices of (1, 1) and (1, -5), you can see that the midpoint between them is (1, -2).
What are the coordinates of the center of the hyperbolic inequality 
?
What are the coordinates of the center of the hyperbolic inequality 
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The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. In both, (h, v) is the center of the hyperbola. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. Because the two terms are combined using subtraction and the x-term appears first, this inequality represents a horizontal hyperbola. To derive the center of a hyperbola from its equation or inequality, flip the sign of the constants that appear after the x and y in the equation or inequality. The constant following x is 3, so the x-coordinate of the center is -3. No constant follows y, so the y-coordinate of the center is 0.
The graph of the hyperbolic inequality 
appears as follows:
As you can see, the midpoint between the two vertices of (-6, 0) and (0, 0) is (-3, 0).
The equation for a horizontal hyperbola is 
The graph of the hyperbolic inequality 
As you can see, the midpoint between the two vertices of (-6, 0) and (0, 0) is (-3, 0).
Which equation does this graph represent?
Which equation does this graph represent?
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The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. In both, (h, v) is the center of the hyperbola. The graph shows a horizontal hyperbola, so in its corresponding equation the x-term must appear first. The center lies at (-2, -1), so x must be followed by the constant 2, and y must be followed by the constant 1. The graph shows a hyperbola rather than an ellipse, so the x- and y-terms must be combined using subtraction rather than addition.
The equation for a horizontal hyperbola is 
