Question 123608
The data in the table shows the height in metres above the ground of a snowball at various times. Showing all work algebraically, create a function that represents the height of the snowball after t seconds.
TIME(sec)--- 1,2,3,4,5
HEIGHT(m)---25.1,38.4,41.9,35.6,19.5
Can someone please help me?
<pre><font face = "book antiqua" size = 5><b>

Although it isn't necessary, it is instructive to
first plot the 5 points with the times on the x-axis
and the heights on the y-axis:

(1,25.1), (2,38.4), (3,41.9), (4,35.6), (5,19.5)

{{{ drawing(400,600,-2,6,-2,43,
graph(400,600,-2,6,-2,43),
locate(1-.061,25.1+.7,o),locate(2-.061,38.4+.7,o),locate(3-.061,41.9+.7,o),locate(4-.061,35.6+.7,o),locate(5-.061,19.5+.7,o)
 )}}} 

It looks as though a parabola would pass through all those points,
so sketch one which passes through all 5 of those points
like this:

{{{ drawing(400,600,-2,6,-2,43,
graph(400,600,-2,6,-2,43,-4.9x^2+28x+2),
locate(1-.061,25.1+.7,o),locate(2-.061,38.4+.7,o),locate(3-.061,41.9+.7,o),locate(4-.061,35.6+.7,o),locate(5-.061,19.5+.7,o)
 )}}} 

Now we know that the equation of the height h(t) in meters above the
ground of a projectile is given by this formula:

h(t) = h<sub>0</sub> + v<sub>0</sub>t - 4.9t²

where h<sub>0</sub> = the original (initial) height above the
ground that the projectile was thrown, and
v<sub>0</sub> = the original (initial) speed in m/s at which
the projectile was thrown upward.

So we substitute the first two points. (1,25.1) and (2,38.4): 

Substituting t=1 and h(1) = 25.1, we have the equation

h(1) = h<sub>0</sub> + v<sub>0</sub>(1) - 4.9(1)²
25.1 = h<sub>0</sub> + v<sub>0</sub> - 4.9
  30 = h<sub>0</sub> + v<sub>0</sub>

Substituting t=2 and h(2) = 38.4, we have the equation

h(2) = h<sub>0</sub> + v<sub>0</sub>(2) - 4.9(2)²
38.4 = h<sub>0</sub> + 2v<sub>0</sub> - 4.9(4)
38.4 = h<sub>0</sub> + 2v<sub>0</sub> - 19.6
  58 = h<sub>0</sub> + 2v<sub>0</sub>

So we have this system of two equations in two
unknowns:

  30 = h<sub>0</sub> +  v<sub>0</sub>
  58 = h<sub>0</sub> + 2v<sub>0</sub>
 
Can you solve that system for h<sub>0</sub> and v<sub>0</sub>?
If not post again asking how.

Answer h<sub>0</sub> = 2 meters, v<sub>0</sub> = 28 m/s

So substituting in:

h(t) = h<sub>0</sub> + v<sub>0</sub>t - 4.9t²

we find the function h(t) is

h(t) = 2 + 28t - 4.9t²

So really all you needed were two points.
The other three were not necessary, but they
helped make the problem clearer.

Edwin</pre>