This parabola can be drawn in several ways:

**x = y²/4**, or **y² = 4x**, or parametrically **x = t²/4, y = t**

and the directrix **x = -1**. Enter a point at the focus **(1, 0)** , and another on the curve, labelled **‘A’**. Construct a tangent through A. construct a straight line through A and the Focus, and find its other intersection (‘B’) with the parabola (use ‘Add point’ with Ctrl). Double-click on ‘B’ to establish its association with the parabola, then draw a tangent through ‘B’. Find the intersection of the two tangents, and display the angle (make sure you are in degrees).

Move ‘A’ around to domstrate that the two tangents meet on the directrix at 90°.

CAREFUL: If the second intersection, ‘B’, goes off the page, it can get lost and the construction collapses. If that happens there is always CTRL-Z (Command Z on Mac) to get you out of jail! (Undo)

Download Autograph files: parabola (parametric), and parabola2 (cartesian)

In each file, an ANIMATION is set up to demonstrate this.

The best way to explore this is to use paramteric equations. First make sure you are set to DEGREES (not radians), and in 2D, enter **x = (vcosb)t, y = h + (vsinb)t − ½gt²** with startup options on “Manual”, with ‘t’ from 0 to 8 (sec), step 0.01. Set the constants to: ‘v’ = 10 (m/s), ‘b’ = 45 (°). h = 1.8 (m), g = 9.81 (m/s²).

After plotting, select the curve and right-click “enter point on curve” at t = 0, with t-snap = 0.1, and plot teh Velocity Vector.

For 3D, enter** x = (vcosb)(cosa)t, y =(vcosb)(sina)t, z=h+(vsinb)t − ½gt²** where ‘b’ is the angle to the horizontal (as in 2D), and ‘a’ is the angle round from the ‘x’ axis. Proceed as with 2D to place a point at t = 0 and a velocity vector. Draw the plane **z = 0** to represent the ground, and a number of circles on the grass using the form **x = 10cost, y = 10sint, z = 0**.

To investigate, either in 2D or 3D, note that the maximum orizontal range ooccuras at 45°, whereas the maximum range on the ground occurs around 39°. This analysis assumes that the javelin can be thrown at the same speed at all angles, which is clearly not the case. You can show that the range is much more affected by the initial velocity of throw than by the angle. Research elswhere suggests that the physiology of the human arm allows the maximum speed of projection at around 31°, which is the angle that most javelin throwers use.

Download **Autograph files**: javelin-2D – javelin -3D

Just enter “y = ” then a string of expressions, each separated by a comma, eg:

y = −4, 2x, 0.4x(6 − x), 2

then in the “start-up options” enter boundary values for ‘x’, eg in this example

x: -4, −2, 1, 5, 8

Download Autograph File “piecewise“