This is an applet to explore the equation of a parabola and its properties. The equation used is the standard equation that has the form(y - k)2 = 4a(x - h)
where h and k are the x- and y-coordinates of the vertex of the parabola and a is a non zero real number (in this investigation we consider only cases with positive a). For the definition and construction of a parabola Go here.
Examples of applications of the parabolic shape as Parabolic Reflectors and Antennas and a tutorial on how to Find The Focus of Parabolic Dish Antennas and on How Parabolic Dish Antennas work? are included in this site.
The exploration is carried out by changing the parameters h, k and a included in the above equation. Follow the steps in the tutorial below.
For similar tutorials on circle , Ellipse and the hyperbola can be found in this site.
Answers and Solutions to questions 2 to 9 can be found here
1 - click on the button above "click here to start" and MAXIMIZE the window obtained. At the start a = 1, h = 0 and k = 0.
2 - Keep the values of a, h and k as above (do not change the positions of the sliders). Find the equation of the directrix and the coordinates of the vertex V and focus F. Find the equation of the axis of symmetry of the parabola (line through V and F).
3 - Use the top slider to set a = 2 and answer the same questions as in part 2 above.
4 - Set a = 1, h = 0 and change k (using the slider). Find a relationship between the y-coordinate of F and parameter k. Find a relationship between the y-coordinate of V and k. Find a relationship between the position (or equation) of the axis of the parabola and k. Does the position of the vertex change?
5 - Set a = 1, k = 0 and change h (using the slider). Find a relationship between the x-coordinate of F and parameter h. Find a relationship between the x-coordinate of V and h. Find a relationship between the position (or equation) of the directrix of the parabola and h. Does the position of the axis change?
6 - Use parts 1,2,3,4 and 5 above to find the coordinates of V and F and the equations of the directrix and axis of the parabola in terms of h and k.
7 - Set a = 1, k = 0 and change h. Which values of h give two y-intercepts? Which values of h give no y-intercepts? Which values of h give one y-intercept?Explain your answers analytically.(Hint: find the y-intercepts by setting x = 0 and solve for y).
8 - Investigate the x-intercept. Explain why the parabola as defined above has one x-intercept only.
9 - Exercise: Show that the following equation
y2 - 4y - 4x = 0
can be written as
(y-k)2 = 4a(x - h)
Hint: put all terms with y and y2 together in one side and all terms with x in the other side of the equation. Complete the square for the expression containing y and y2.
Find a, h and k. Find the coordinates of V and F. Find the equations of the axis and directrix of this parabola. Put the values of a, h and k in the applet and check your answer.
If needed, Free graph paper is available.
More references and links to topics related to the parabola
Tutorial on How Parabolic Dish Antennas work?
Tutorial on how to Find The Focus of Parabolic Dish Antennas.
Use of parabolic shapes as Parabolic Reflectors and Antannas.
Interactive tutorial on how to find the equation of a parabola.
Define and Construct a Parabola.