Determine the equation of a parabola and graph a parabola - Pre-Calculus
Card 0 of 184
Which is the equation for a parabola that opens down?
Which is the equation for a parabola that opens down?
The answer is
because it is the only degree-2 polynomial with a negative leading coefficient.
The answer is because it is the only degree-2 polynomial with a negative leading coefficient.
Compare your answer with the correct one above
Find the directerix for the parabola with the following equation:

Find the directerix for the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the directerix of the parabola with the following equation:

Find the directerix of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
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:
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 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.
Compare your answer with the correct one above
Express the following equation for a parabola in standard form:

Express the following equation for a 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:



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:
Compare your answer with the correct one above
Which direction does the parabola open?

Which direction does the parabola open?
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.
Compare your answer with the correct one above
Determine the direction in which the following parabola opens.

Determine the direction in which the following parabola opens.
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.
Compare your answer with the correct one above
Determine what direction the following parabola opens: 
Determine what direction the following parabola opens:
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 term determines whether if the parabola opens upward or downward. Since the
term in the function
is
, the parabola will open downward.
Compare your answer with the correct one above
Determine what direction will the following function open: 
Determine what direction will the following function open:
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 in its standard form for parabolas, which is
.
Regroup the terms.
Since the coefficient of the term is negative, the parabola will open downward.
Compare your answer with the correct one above
Find the focus and the directrix of the following parabola:
.
Find the focus and the directrix of the following parabola: .
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 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 .
Compare your answer with the correct one above
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 , and the y-intercept is
, find the equation of the parabola, if possible.
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 , 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.
Compare your answer with the correct one above
Rewrite
in standard form.
Rewrite in standard form.
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 , and simplify.
Compare your answer with the correct one above
Find the directerix of the parabola with the following equation:

Find the directerix of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the directerix of the parabola with the following equation:

Find the directerix of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the focus of the parabola with the following equation:

Find the focus of the parabola with the following equation:
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:
, 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
Compare your answer with the correct one above
Find the focus of the parabola with the following equation:

Find the focus of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the focus of the parabola with the following equation:

Find the focus of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the directerix of the parabola with the following equation:

Find the directerix of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the directerix of the parabola with the following equation:

Find the directerix of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above
Find the focus of the parabola with the following equation:

Find the focus of the parabola with the following equation:
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:
, 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
.
Compare your answer with the correct one above