Quadratic Function Model

A ball is tossed upward. Its height after 't' seconds is given in the following ordered pairs. x values are the seconds, y values are the height in feet. (0.5,26.5), (1,39.5), (1.5,44.5), (2,41.5), (2.5,30.5)

Find a quadratic function to model the data. Use the model to determine when the ball reaches its maximum height, as well as the value of the maximum height.

I just need help finding the quadratic function. I'm pretty much clueless when it comes to that part. I just need to find the function and need to know what to do with it, then I can solve the problem.

Your generic quadratic looks like y = ax^2 + bx + c.

Then simply substitute in points to get a series of equations



{eq1 (1,39.5)} 39.5 = a+b+c
{eq2 (2,41.5)} 41.5 = 4a + 2b + c
{eq3 (0.5, 26.5)} 26.5 = a/4 + b/2 + c
Now you have a system of 3 equations

Now you just need to be systematic on solving this system

eq2 - eq1 --> 2 = 3a + b {eq4}
eq2 - eq3 --> 15 = 15/4 a + 3/2 b {eq5}

Now solve eq4 for b
b = 2-3a

Now sub this into eq5 for b

15 = 15/4 a + 3/2 (2-3a)

Multiply through by 4

60 = 15a + 6(2-3a) = 15a + 12-18a = 12-3a

48 = -3a

a = -16

b = 2-3a, so b = 2-3(-16) = 2+48 = 50

39.5 = a+b+c

39.5 = -16 + 50 + c

39.5 = 34 + c

5.5 = c

Then the quadratic is y = -16x^2 + 50x + 5.5