We begin our study of the conic sections with parabolas, since we have already seen parabolas described as graphs of quadratic functions, \(f(x) = ax^2 + bx + c\) (\(a \neq 0\)). It turns out that we can also describe parabolas in terms of distances.

In the GeoGebra interactive below, click and drag the focus, \(F\) up and down to generate different parabolas. Click and drag the point \(A\) along the parabola. Each dashed line from the point \(F\) to a point on the curve has the same length as the dashed line from the point on the curve to the line \(D\).

The point suggestively labeled \(V\) is, as you may expect, the vertex. The vertex is the point on the parabola closest to the focus.

We want to use only the distance definition of parabola to derive the equation of a parabola and, if all is right with the universe, we should get an expression much like those studied in Section QuadraticFunctions.

For simplicity, assume that the vertex is \((0,0)\) and that the parabola opens upwards. Let \(p\) denote the directed distance from the vertex to the focus, which by definition is the same as the distance from the vertex to the directrix. Hence, the focus is \((0,p)\) and the directrix is the line \(y = -p\). Our picture becomes

From the definition of parabola, we know the distance from \((0,p)\) to \((x,y)\) is the same as the distance from \((x,-p)\) to \((x,y)\). Using the Distance Formula, Equation distanceformula, we get

\[ \begin{array}{rclr} \sqrt {(x -0)^2 + (y-p)^2} & = & \sqrt {(x-x)^2 + (y - (-p))^2} & \\ \sqrt {x^2 + (y-p)^2} & = & \sqrt {(y+p)^2} & \\ x^2 + (y-p)^2 & = & (y+p)^2 & \mbox {square both sides} \\ x^2 + y^2 - 2py + p^2 & = & y^2 + 2py + p^2 & \mbox {expand quantities} \\ x^2 & = & 4py & \mbox {gather like terms} \\ \end{array} \]

Solving for \(y\) yields \(y = \frac {x^2}{4p} = \frac {1}{4p} x^2\), which is a quadratic function of the form found in Equation vertexofquadraticfunctions with \(a = \frac {1}{4p}\) and vertex \((0, 0)\).

We know from previous experience that if the coefficient of \(x^2\) is negative, the parabola opens downwards. In the equation \(y = \frac {1}{4p} x^2\) this happens when \(p < 0\). In our formulation, we say that \(p\) is a ‘directed distance’ from the vertex to the focus: if \(p > 0\), the focus is above the vertex; if \(p < 0\), the focus is below the vertex. The focal length of a parabola, that is, the length from the vertex to the focus, is therefore \(|p|\).

If we choose to place the vertex at an arbitrary point \((h,k)\), we arrive at the following formula using either transformations from Section Transformations or re-deriving the formula from Definition paraboladefn.

Notice that in the standard equation of the parabola above, only one of the variables, \(x\), is squared. As we’ll see in the coming sections, this is a quick way to distinguish the equation of a parabola from equations representing the other conic sections.

Before embarking on an example, we take a moment to better illustrate the affect of the focal length \(|p|\) on the graph of a parabola. In the GeoGebra interactive below, adjust the slider for \(p\) and observe the resulting parabola with focus \(F\). In general, as the focal length \(|p|\) increases, the parabola becomes wider.

The dashed line segment in the interactive above is called the latus rectum of the parabola. More specifically, the latus rectum of a parabola is the line segment with endpoints on the parabola which contains the focus and is parallel to the directrix.

We leave it to the reader to show that the length of the latus rectum, called the focal diameter of the parabola is \(|4p| = 4|p|\), which appears ever so conveniently in the standard form as stated in Equation standardvparabola.

Knowing the focus and focal diameter allows to plot two points on the parabola in addition to the vertex, thus producing a more accurate graph.

We can produce ‘horizontal’ parabolas in the \(xy\)-plane by reflecting our so-called ‘vertical’ parabolas about the line \(y=x\). As you may recall from Section InverseFunctions, we accomplish this algebraically by interchanging the variables \(x\) and \(y\). Such parabolas necessarily open to the left or to the right, which means that unlike the vertical parabolas, these parabolas do not represent \(y\) as a function of \(x\). As we shall see, however, they can implicitly describe \(y\) as a function of \(x\), provided certain restrictions are in place.

As we saw in Section InverseFunctions, when we reflect a horizontal line across the line \(y=x\), we obtain a vertical line, and, as a result, the directrix of a horizontal parabola is a vertical line. Moreover, the focus of a horizontal parabola is either to the left or right of the directrix.

In the GeoGebra interactive below adjust the slider for the focal length, \(p\) and observe the resuting ‘horizontal’ parabola.

As we have seen, not all equations which describe parabolas will immediately match Equation standardvparabola or Equation standardhparabola. Indeed, completing the square as we did with the equation in number ctsparabolaex in Example horizontalparabolaex will be a necessary skill not only in this section, but in the rest of this chapter.

For parabolas, we summarize the procedure for putting an equation of a parabola into standard form below. Of key importance is that in the equation for a parabola, one, and only one, of the variables are squared.

To Write the Equation of a Parabola in Standard Form
  1. Group the variable which is squared on one side of the equation and position the non-squared variable and the constant on the other side.
  2. Complete the square if necessary and divide by the coefficient of the perfect square.
  3. Factor out the coefficient of the non-squared variable from it and the constant.

In studying quadratic functions, we have seen parabolas used to model physical phenomena such as the trajectories of projectiles. Other applications of the parabola concern its ‘reflective property’ which necessitates knowing about the focus of a parabola. For example, many satellite dishes are formed in the shape of a paraboloid of revolution as depicted below.

Every cross section through the vertex of the paraboloid is a parabola with the same focus. To see why this is important, imagine the dashed lines below as electromagnetic waves heading towards a parabolic dish. It turns out that the waves reflect off the parabola and concentrate at the focus which then becomes the optimal place for the receiver.

If, on the other hand, we imagine the dashed lines as emanating from the focus, we see that the waves are reflected off the parabola in a coherent fashion as in the case in a flashlight. Here, the bulb is placed at the focus and the light rays are reflected off a parabolic mirror to give directional light.

In the GeoGebra interactive below, click and drag the point on the parabola \(A\) to see how a rays from a light source at the focus \(F\) reflect off a parabolic mirror. Additionally, clicking and dragging the focus \(F\) allows us to observe this phenomenon on a variety of parabolas.