Lesson 4

Parabolas

In this lesson students will become familiar with the equations and graphs of parabolas. The definition of a parabola will be learned both algebraically and using the distance relationship. Students will learn how to construct a parabola using Geometer’s Sketchpad and how to prove that this construction is a parabola. Applications of parabolas will be explored.

Defintion: A parabola is the set of all points P(x, y) in the plane whose distance to a fixed point, called the focus, equals its distance to a fixed line, called the directrix.

Some other important information includes the following:

• The vertex of the parabola is located at (h, k).
• The focus of the parabola is located inside the parabola a distance of p from the vertex.
• The line called the directrix is located a distance of p from the vertex on the outside of the parabola.

• The axis of symmetry is x=h, for horizontal directrix parabola, and y=k, for vertical directrix parabola. Standard equations for the ellipse and the graph of each are shown below.
• Standard equations for the parabola and the graph of each are shown below.

Note that when the vertex of the parabola is translated to a point (h, k) so that the vertex is other than the origin, the equation will become (x-h)²=4p(y-k) for the parabolas shown in (a) and (b), and will become (y-k)²=4p(x-h) for (c) and (d).

Try constructing a parabola using GSP. First complete the paper folding activity by clicking here. Use the instructions provided to complete this construction by clicking here. The parabola that has been constructed is not necessarily oriented to the standard coordinate system. After your construction is complete, prove that the construction is a parabola. Show your proof using two different methods, geometric and algebraic. Click here to explore the construction of a parabola on GSP.

Practice problem 1: Suppose that a golf ball travels a distance of 600 feet as measured along the ground and reaches an altitude of 200 feet. If the origin represents the tee and the ball travels along a parabolic path that opens downward, find an equation for the path of the golf ball.

Practice problem 2: A parabola defined by the equation 4x+y²-6y=9 is translated 2 units up and 4 units to the left. Write the standard equation of the resulting parabola.