What is the general and standard form of these equations ?

Question

Some me children are playing pirate ships. They made catapults that they use with foam balls to attack their enemies. The two ships are 4m apart. Considering the height of the catapult to be zero, here are the results of the first 3 launches. The first launch is too short by 1m and an observer notes that after travelling a distance of 2m, the ball had reached a height of 1m. The second ball reached the mast of the ship at a height of 0.5m and fell to the ground 0.5m beyond the target. During the third launch, an observer notes that the ball reached a maximum height of 3m after traveling a distance of 3m.

a) Determine the standard form of the equation representing the path of the first ball launched( distance as a function of the height)

b) determine the general form of the equation representing the path of the second ball launched.

c) determine the standard form of the equation representing the path of the third ball launched.

Answer 1

The task here is to take three points and fit them to a parabola. Fortunately three points is all that we require!

 

(a) The points here are (0,0), (2,1), and (3,0). Since the first and third points are on the x-axis, the best angle here is to start with the factored form.

 

y = a(x - x1)(x - x2)

y = a x (x - 3)

 

Now we use the second point to find the coefficient a. 

 

1 = a • 2 • (2 - 3)

a = -½

 

Thus y = -½x(x - 3)

 

To put it in standard form (by which I assume they mean vertex or h-k form), bring the x inside the parenthetical and complete the square; don't forget to multiply the extra constant by the -½ and add outside. 

 

I get y = -½(x - 3/2)² + 9/8.

 

(b) The points are now (0,0), (4, ½), and (9/2,0). Again they gave us two roots so the approach is the same as before:

 

y = a(x)(x - 9/2)

½ = a • 4 (4 - 4½)

a = -¼

 

For this one they ask for the general form. I assume this means with coefficients a, b, and c: thus y = -¼x² + 9/8 x.

 

(c) This time we only have two points, (0,0), (3,3). This would be a problem but for the fact that the second point is identified as the vertex. So we can set the standard (h-k) form directly, then use the first point to again find a:

 

y = a(x - 3)² + 3

0 = a(0-3)² + 3

a = -1/3

 

So y = -1/3 (x - 3)² + 3.