Graphing Hyperbolic Functions - Algebra II
Card 0 of 4
In which direction does the graph of the above hyperbola open?
In which direction does the graph of the above hyperbola open?
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.
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What are the vertices of this hyperbola?
What are the vertices of this hyperbola?
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
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Write the expression for this hyperbola in standard form:
Write the expression for this hyperbola in standard form:
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
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Find the vertices of the following hyperbolic function:
Find the vertices of the following hyperbolic function:
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
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