Geometry Worksheets

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Parabola

By Mandi Elam

Parabolas are conic sections. They are defined as the collection of all points in a plane that are equidistant to the focus and the directrix.

Why is this concept useful?

Parabolas are useful in a myriad of real world applications. Parabolic shapes show up in technology applications often, from satellites to photography reflectors to science applications such as gravity versus time.

Where does this concept fit into the curriculum?

Geometry

How can we use the concept:

Parabolas show up in mathematics in several areas. In Algebra, the parabola shows up as the shape of the graph for the function family f(x)=x2. We look at transformations of this graph and explore both the standard and vertex form of quadratic functions to describe those transformations.

Standard Form of a Quadratic:

f(x)=ax2+bc+cf( x) =ax^{2} +bc+c

Vertex Form of a Quadratic:

f(x)=a(x−h)2+kf( x) =a( x-h)^{2} +k

Parabolas are interesting in that they can also be defined in terms of distance. A parabola is the set of all points in a plane that are equidistant (the same distance) from the focus and the directrix.

In this definition, the focus is a fixed point and the directrix is a fixed line that is perpendicular to the axis of symmetry of the parabola. The midpoint between the focus and directrix is the vertex. The equations of parabolas use parameter

p, where |p| is the distance from the vertex to both the directrix and the focus.

Another difference is in orientation. When dealing with functions, the parabola could only be open upward or downward. However, parabolas may open either left or right with a horizontal axis of symmetry.


Sample Math Problems

Example 1: Find the Equation of a Parabola with the Distance Formula
Given focus F(2, 0) and directrix x=-2.
In the definition of a parabola, it is stated that all points on the parabola are equidistant from the directrix and the focus. So some point P(x,y) is equidistant from F(2,0)and perpendicular to point D(-2,y).This means line segment PD and line segment PF have the same length.

Thus PD=PF, and we can do the following:

Example 2: Find the Standard Form of the Parabola

Given the graph, find the Standard form of the parabola.

Solution:

Example 3: Find the Standard Form of the Parabola Given V(0,0)and F(-1.5,0)

Solution:


The standard form can be found with just the vertex and either the directrix or focus because both of those are the same distance p from any point on the parabola. So the same method can be used given the directrix instead of the focus.

Horizontal Parabola, because focus is F(-1.5, 0), so axis of symmetry is y=0

p=-1.5since F(p,0); (h,k)=(0, 0)

Simplify to get the equation of the parabola

Example 4: Graphing Parabolas

Graph the parabola

x+3=−136(y−7)2x+3=-\frac{1}{36}( y-7)^{2}

From the form, we can tell that the graph is a horizontal parabola opening left.

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Practice Math Problems

1. Use the distance formula to find the equation of a parabola with focus F(-5,0)and directrix x=5.

2. Write the equation in standard form for a parabola with vertex V(0,0) and focus F(6,0)

3. Given the following equation, F(1.5,0)find the following and graph the parabola:

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