Parabolas - Pre-Calculus
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Determine the direction in which the following parabola opens, if the y-axis is vertical and the x-axis is horizontal:
Determine the direction in which the following parabola opens, if the y-axis is vertical and the x-axis is horizontal:
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In order to determine which direction the parabola opens, we must first put the equation in standard form, which can be expressed in one of the following two ways:
If the equation is for 
as in the first above, the parabola opens up if 
is positive and down if 
is negative. If the equation is for 
as in the second above, the parabola opens right if 
is positive and left if 
is negative. Rearranging our equation, we get:
We can see that our equation is for 
, which means the parabola will open either left or right. The sign of the first term is negative, so this parabola will open to the left.
In order to determine which direction the parabola opens, we must first put the equation in standard form, which can be expressed in one of the following two ways:
If the equation is for
We can see that our equation is for
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Express the following equation for a parabola in standard form:
Express the following equation for a parabola in standard form:
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In order to be in standard form, the equation for a parabola must be written in one of the following ways:
We can see that our equation has all of the components of the first form above, so now all we must do is some algebra to rearrange the equation and express the function y in terms of x. We start by simplifying the fraction on the left side of the equation, and then we isolate y to give us the equation of the parabola in standard form:
In order to be in standard form, the equation for a parabola must be written in one of the following ways:
We can see that our equation has all of the components of the first form above, so now all we must do is some algebra to rearrange the equation and express the function y in terms of x. We start by simplifying the fraction on the left side of the equation, and then we isolate y to give us the equation of the parabola in standard form:
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Which direction does the parabola open?
Which direction does the parabola open?
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For the function
The parabola opens upwards if a>0
and downards for a<0
Because
The parabola opens upwards.
For the function
The parabola opens upwards if a>0
and downards for a<0
Because
The parabola opens upwards.
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Determine the direction in which the following parabola opens.
Determine the direction in which the following parabola opens.
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For the function
The parabola opens upwards if a>0
and downards for a<0
Because
The parabola opens downwards.
For the function
The parabola opens upwards if a>0
and downards for a<0
Because
The parabola opens downwards.
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Determine what direction the following parabola opens: 
Determine what direction the following parabola opens:
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The standard form for a parabola is in the form: 
The coefficient of the 
term determines whether if the parabola opens upward or downward. Since the 
term in the function 
is 
, the parabola will open downward.
The standard form for a parabola is in the form:
The coefficient of the
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Determine what direction will the following function open: 
Determine what direction will the following function open:
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Use the FOIL method to determine 
in its standard form for parabolas, which is 
.
Regroup the terms.
Since the coefficient of the 
term is negative, the parabola will open downward.
Use the FOIL method to determine
Regroup the terms.
Since the coefficient of the
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Find the focus and the directrix of the following parabola: 
.
Find the focus and the directrix of the following parabola:
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To find the focus from the equation of a parabola, first set the equation to resemble the form 
where 
represents any numerical value.
For our problem, it is already in this form.
Therefore,
.
Solve for 
then 
.
The focus for this parabola is given by 
.
So, 
is the focus of the parabola.
The directrix is represented as 
.
Therefore, the directrix for this problem is 
.
To find the focus from the equation of a parabola, first set the equation to resemble the form
For our problem, it is already in this form.
Therefore,
Solve for
The focus for this parabola is given by
So,
The directrix is represented as
Therefore, the directrix for this problem is
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If the vertex of the parabola is 
, and the y-intercept is 
, find the equation of the parabola, if possible.
If the vertex of the parabola is
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First, write the equation of the parabola in standard form.
Determine the values of the coefficients. The value of the y-intercept is 4, which means that 
.
Write the vertex formula.
The given point of the vertex is 
, which indicates that:
Substitute the value of the point and 
into the standard form.
Substitute this value into 
to determine the value of 
.
Substitute the values of 
coefficients into the standard form of the parabola.
First, write the equation of the parabola in standard form.
Determine the values of the coefficients. The value of the y-intercept is 4, which means that
Write the vertex formula.
The given point of the vertex is
Substitute the value of the point and
Substitute this value into
Substitute the values of
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Rewrite 
in standard form.
Rewrite
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The standard form of a parabola is 
.
Factorize the right side of 
, and simplify.
The standard form of a parabola is
Factorize the right side of
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Find the directerix of the parabola with the following equation:
Find the directerix of the parabola with the following equation:
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Recall the standard form of the equation of a vertical parabola:
, where 
is the vertex of the parabola and 
gives the focal length.
When 
, the parabola will open up.
When 
, the parabola will open down.
For the parabola in question, the vertex is 
and 
. This parabola will open up. Because the parabola will open up, the directerix will be located 
units down from the vertex. The equation for the directerix is then 
.
Recall the standard form of the equation of a vertical parabola:
When
When
For the parabola in question, the vertex is
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Find the directerix for the parabola with the following equation:
Find the directerix for the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a vertical parabola:
, where 
is the vertex of the parabola and 
gives the focal length.
When 
, the parabola will open up.
When 
, the parabola will open down.
For the parabola in question, the vertex is 
and 
. This parabola will open up. Because the parabola will open up, the directerix will be located 
unit down from the vertex. The equation for the directerix is then 
.
Recall the standard form of the equation of a vertical parabola:
When
When
For the parabola in question, the vertex is
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Find the directerix of the parabola with the following equation:
Find the directerix of the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a vertical parabola:
, where 
is the vertex of the parabola and 
gives the focal length.
When 
, the parabola will open up.
When 
, the parabola will open down.
Start by putting the equation in th estandard form of the equation of a vertical parabola.
Isolate the 
terms to one side.
Complete the square for the 
terms. Remember to add the same amount on both sides!
Factor out both sides of the equation to get the standard form of a vertical parabola.
For the parabola in question, the vertex is 
and 
. This parabola will open down. Because the parabola will open down, the directerix will be located 
units above the vertex. The equation for the directerix is then 
.
Recall the standard form of the equation of a vertical parabola:
When
When
Start by putting the equation in th estandard form of the equation of a vertical parabola.
Isolate the
Complete the square for the
Factor out both sides of the equation to get the standard form of a vertical parabola.
For the parabola in question, the vertex is
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Find the focus of the parabola with the following equation:
Find the focus of the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a vertical parabola:
, where 
is the vertex of the parabola and 
gives the focal length.
When 
, the parabola will open up.
When 
, the parabola will open down.
For the parabola in question, the vertex is 
and 
. This parabola will open down. Because the parabola will open down, the focus will be located 
units down from the vertex. The focus is then located at 
Recall the standard form of the equation of a vertical parabola:
When
When
For the parabola in question, the vertex is
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Find the focus of the parabola with the following equation:
Find the focus of the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a vertical parabola:
, where 
is the vertex of the parabola and 
gives the focal length.
When 
, the parabola will open up.
When 
, the parabola will open down.
For the parabola in question, the vertex is 
and 
. This parabola will open down. Because the parabola will open down, the focus will be located 
units down from the vertex. The focus is then located at 
.
Recall the standard form of the equation of a vertical parabola:
When
When
For the parabola in question, the vertex is
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Find the focus of the parabola with the following equation:
Find the focus of the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a vertical parabola:
, where 
is the vertex of the parabola and 
gives the focal length.
When 
, the parabola will open up.
When 
, the parabola will open down.
Start by putting the equation into the standard form.
Isolate the 
terms on one side.
Complete the square. Remember to add teh same amount on both sides!
Factor both sides of the equation to get the standard equation for the parabola.
For the parabola in question, the vertex is 
and 
. This parabola will open up. Because the parabola will open up, the focus will be located 
unit up from the vertex. The focus is then located at 
.
Recall the standard form of the equation of a vertical parabola:
When
When
Start by putting the equation into the standard form.
Isolate the
Complete the square. Remember to add teh same amount on both sides!
Factor both sides of the equation to get the standard equation for the parabola.
For the parabola in question, the vertex is
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Find the directerix of the parabola with the following equation:
Find the directerix of the parabola with the following equation:
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Recall the standard form of the equation of a horizontal parabola:
, where 
is the vertex of the parabola and 
is the focal length.
When 
, the parabola opens to the right.
When 
, the parabola opens to the left.
For the given parabola, the vertex is 
and 
. This means the parabola is opening to the left and that the directerix will be located 
unit to the right of the vertex. The directerix is then 
.
Recall the standard form of the equation of a horizontal parabola:
When
When
For the given parabola, the vertex is
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Find the directerix of the parabola with the following equation:
Find the directerix of the parabola with the following equation:
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Recall the standard form of the equation of a horizontal parabola:
, where 
is the vertex of the parabola and 
is the focal length.
When 
, the parabola opens to the right.
When 
, the parabola opens to the left.
For the given parabola, the vertex is 
and 
. This means the parabola is opening to the left and that the directerix will be located 
units to the right of the vertex. The directerix is then 
.
Recall the standard form of the equation of a horizontal parabola:
When
When
For the given parabola, the vertex is
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Find the directerix of the parabola with the following equation:
Find the directerix of the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a horizontal parabola:
, where 
is the vertex of the parabola and 
is the focal length.
When 
, the parabola opens to the right.
When 
, the parabola opens to the left.
Put the given equation into the standard form. Start by isolating the 
terms to one side.
Complete the square. Remember to add the same amount to both sides of the equation.
Factor both sides of the equation to get the standard form of the equation of a horizontal parabola.
For the given parabola, the vertex is 
and 
. This means the parabola is opening to the right and that the directerix will be located 
units to the left of the vertex. The directerix is then 
.
Recall the standard form of the equation of a horizontal parabola:
When
When
Put the given equation into the standard form. Start by isolating the
Complete the square. Remember to add the same amount to both sides of the equation.
Factor both sides of the equation to get the standard form of the equation of a horizontal parabola.
For the given parabola, the vertex is
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Find the focus of the parabola with the following equation:
Find the focus of the parabola with the following equation:
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Recall the standard form of the equation of a horizontal parabola:
, where 
is the vertex of the parabola and 
is the focal length.
When 
, the parabola opens to the right.
When 
, the parabola opens to the left.
For the given parabola, the vertex is 
and 
. This means the parabola is opening to the right and that the focus will be located 
units to the right of the vertex. The focus is then located at 
.
Recall the standard form of the equation of a horizontal parabola:
When
When
For the given parabola, the vertex is
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Find the focus of the parabola with the following equation:
Find the focus of the parabola with the following equation:
Tap to reveal answer
Recall the standard form of the equation of a horizontal parabola:
, where 
is the vertex of the parabola and 
is the focal length.
When 
, the parabola opens to the right.
When 
, the parabola opens to the left.
For the given parabola, the vertex is 
and 
. This means the parabola is opening to the right and that the focus will be located 
units to the right of the vertex. The focus is then located at 
.
Recall the standard form of the equation of a horizontal parabola:
When
When
For the given parabola, the vertex is
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