In The Ellipse we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola.

Parabola

Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points [latex]\left(x,y\right)[/latex] in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.

We previously learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.

The line segment that passes through the focus and is parallel to the directrix is called the latus rectum, also called the focal diameter. The endpoints of the focal diameter lie on the curve. By definition, the distance [latex]d[/latex] from the focus to any point [latex]P[/latex] on the parabola is equal to the distance from [latex]P[/latex] to the directrix.

Key features of the parabola

To work with parabolas in the coordinate plane, we consider two cases: those with a vertex at the origin and those with a vertex at a point other than the origin. We begin with the former.

Let [latex]\left(x,y\right)[/latex] be a point on the parabola with vertex [latex]\left(0,0\right)[/latex], focus [latex]\left(0,p\right)[/latex], and directrix [latex]y= -p[/latex] as shown in Figure 4. The distance [latex]d[/latex] from point [latex]\left(x,y\right)[/latex] to point [latex]\left(x,-p\right)[/latex] on the directrix is the difference of the y-values: [latex]d=y+p[/latex]. The distance from the focus [latex]\left(0,p\right)[/latex] to the point [latex]\left(x,y\right)[/latex] is also equal to [latex]d[/latex] and can be expressed using the distance formula.

[latex]\begin{align}d&=\sqrt{{\left(x - 0\right)}^{2}+{\left(y-p\right)}^{2}} \\ &=\sqrt{{x}^{2}+{\left(y-p\right)}^{2}} \end{align}[/latex]

Set the two expressions for [latex]d[/latex] equal to each other and solve for [latex]y[/latex] to derive the equation of the parabola. We do this because the distance from [latex]\left(x,y\right)[/latex] to [latex]\left(0,p\right)[/latex] equals the distance from [latex]\left(x,y\right)[/latex] to [latex]\left(x, -p\right)[/latex].

[latex]\sqrt{{x}^{2}+{\left(y-p\right)}^{2}}=y+p[/latex]

We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.

[latex]\begin{gathered}{x}^{2}+{\left(y-p\right)}^{2}={\left(y+p\right)}^{2} \\ {x}^{2}+{y}^{2}-2py+{p}^{2}={y}^{2}+2py+{p}^{2}\\ {x}^{2}-2py=2py \\ {x}^{2}=4py\end{gathered}[/latex]

The equations of parabolas with vertex [latex]\left(0,0\right)[/latex] are [latex]{y}^{2}=4px[/latex] when the x-axis is the axis of symmetry and [latex]{x}^{2}=4py[/latex] when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.

A General Note: Standard Forms of Parabolas with Vertex (0, 0)

The table below summarizes the standard features of parabolas with a vertex at the origin.

Axis of Symmetry	Equation	Focus	Directrix	Endpoints of Focal Diameter
x-axis	[latex]{y}^{2}=4px[/latex]	[latex]\left(p,\text{ }0\right)[/latex]	[latex]x=-p[/latex]	[latex]\left(p,\text{ }\pm 2p\right)[/latex]
y-axis	[latex]{x}^{2}=4py[/latex]	[latex]\left(0,\text{ }p\right)[/latex]	[latex]y=-p[/latex]	[latex]\left(\pm 2p,\text{ }p\right)[/latex]

(a) When [latex]p>0[/latex] and the axis of symmetry is the x-axis, the parabola opens right. (b) When [latex]p<0[/latex] and the axis of symmetry is the x-axis, the parabola opens left. (c) When [latex]p<0[/latex] and the axis of symmetry is the y-axis, the parabola opens up. (d) When [latex]\text{ }p<0\text{ }[/latex] and the axis of symmetry is the y-axis, the parabola opens down.

The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and focal diameter. When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.

A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the focal diameter, these lines intersect on the axis of symmetry.

Writing Equations of Parabolas in Standard Form

In the previous examples we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

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