Parabolas are defined as conic sections that are formed by cutting a cone with a plane that is parallel to one lateral side of the cone. Parabolas are formed by the set of all points that are equidistant with respect to a line, called the directrix, and to a point, called the focus.

Here, we will learn how to define an equation of the parabola. We will look at two different cases: when the vertex of the parabola is located at the origin and when the vertex is located outside the origin. We will apply what we have learned by solving some problems.

## Parabolas that have the vertex at (0, 0)

One way to define parabolas is by using the general equation $latex y={{x}^2}$. This equation represents a parabola with a vertex at the origin, (0, 0), and an axis of symmetry at $latex x=0$.

Additionally, we can also use the focus and directrix of the parabola to obtain an equation since each point on the parabola is equidistant from the focus and directrix.

The vertex of the parabola is located in the middle between the focus and the directrix. The focus is a point that is located on the axis of symmetry, while the directrix is a line that is perpendicular to the axis of symmetry. The following image shows the focus, vertex, and directrix:

We know that we can describe a parabola using the general form $latex y=a{{x}^2}$. It is possible to rewrite this equation using the form $latex {{x}^2}=4py$, where *p* is a constant used to find the focus and directrix. This corresponds to a parabola with a vertical orientation.

On the other hand, when we have a horizontally oriented parabola, we have the equation $latex {{y}^2}=4px$.

The diagram shows us the four different cases that we can have when the parabola has a vertex at (0, 0). When the variable *x* is squared, the parabola is oriented vertically and when the variable *y* is squared, the parabola is oriented horizontally.

Furthermore, when the value of *p* is positive, the parabola opens towards the positive part of the axes, that is, upwards or to the right. On the other hand, when the value of *p* is negative, the parabola opens towards the negative part of the axes, that is, downwards or to the left.

**EXAMPLE 1**

What is the equation of a parabola that has a focus at (0, 3)?

##### Solution

We know that the vertex of the parabola is located at (0, 0). The fact that the focus is on (0, 3) indicates that the value of *p* is the value of *y* and is positive. This means that the parabola will open upwards. Therefore, the general equation is $latex {{x}^2}=4py$.

Substituting 3 for *p*, we have:

$latex {{x}^2}=4(3)y$

$latex {{x}^2}=12y$

**EXAMPLE 2**

If we have the parabola $latex y = \frac{1}{2} {{x}^2}$, what is its focus and directrix?

##### Solution

We find the focus and directrix by solving for $latex {{x}^2}$ and determining the value of p. So, we have:

$latex y=\frac{1}{2}{{x}^2}$

$latex 2y={{x}^2}$

We can form the equation $latex 2 = 4p$ and solve for *p*:

$latex 2=4p$

$latex p=\frac{1}{2}$

Therefore, the focus is $latex (0, \frac{1}{2})$ and the directrix is $latex y=-\frac{1}{2}$.

## Parabolas that have the vertex at (h, k)

Not all parabolas have their vertex at the point (0, 0) since in many cases, parabolas have their vertex at the point (*h, k*) that is located outside the origin.

Above we saw that, when the vertex of the parabola is located at the origin, its equation is $latex {{x}^2} = 4py$ or $latex {{y}^2} = 4px$ depending on its orientation.

If we combine these equations with the vertex form of the parabolas, $latex y = a{{(x-h)}^2} -k$, we can form an equation that applies to cases when the vertex is not at the origin.

Therefore, we start by solving for $latex {{(x-h)}^2}$:

$latex y=a{{(x-h)}^2}-k$

$latex {{(x-h)}^2}=\frac{1}{a}(y-k)$

If we compare this equation with $latex {{x}^2}=4py$, we have $latex 4p=\frac{1}{a}$. Therefore:

$latex {{(x-h)}^2}=4p(y-k)$

This is the equation of a vertically oriented parabola. On the other hand, if the parabola is oriented horizontally, its equation will be $latex {{(y-k)}^2}=4p(x-h)$.

The diagram shows us that when the variable *x* is squared, the parabola is oriented vertically and when the variable *y* is squared, the parabola is oriented horizontally. Also, the value of *p* indicates where the parabola opens.

If *p* is positive, the parabola opens towards the positive part of the axes. If *p* is negative, the parabola opens towards the negative part of the axes.

**EXAMPLE 1**

If the vertex of a parabola is located at (-2, 4) and its directrix is $latex y=7$, what is its equation?

##### Solution

We see that the directrix is a horizontal line, so the parabola is oriented vertically and will open up or down. Furthermore, we also see that the directrix is located above the vertex, so the parabola opens downward and the value of *p* is negative.

We can find the value of *p* using the vertex (*h, k*). The equation for a horizontal directrix is $latex y=k-p$. Therefore, we have:

$latex 7=4-p$

$latex 3=-p$

$latex p=-3$

Using these values, we have:

$latex {{(x-h)}^2}=4p(y-k)$

$latex {{(x-(-2))}^2}=4(-3)(y-4)$

$latex {{(x+2)}^2}=-12(y-4)$

**EXAMPLE 2**

What is the vertex, focus, axis of symmetry, and directrix of the parabola $latex {{(x+3)}^2}=4(y+4)$?

##### Solution

We can compare this equation with the general equation given above. Therefore, the vertex is the point (-3, -4). Also, since the variable *x* is squared, the parabola opens upward.

Now, we form the equation $latex 4p = 4$. Solving for *p*, we have $latex p = 1$. This means that the focus is $latex (-3, -4 + 1) = (- 3, -3)$, the axis of symmetry is $latex x=-3$, and the directrix is $latex y= -3-1 = -4$.

## See also

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