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.
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.