Quadratic Functions and Equations?

Question

Some types of worms have a remarkable capacity to live without moisture. The table shows the number of worms y surviving after x days in one study.

X (days) 0 20 40 80 120 160
Y (worms) 50 48 45 36 20 3

(A) Find a quadratic function in the form
f(x) = a(x-h)^2 + K that models these data.

(B) Solve the quadratic equation f(x) = 0. Do both solutions have real meaning? Explain

Answer 1

f(0) = 50 = a(h^2) + K
f(20) = 48 = a(20-h)^2 + K
f(40) = 45 = a(40-h)^2 + K

Expanding, we get three equations:
1) ah^2 + K = 50
2) ah^2 - 40ah + 400a + K = 48
3) ah^2 - 80ah + 1600a + K = 45

Subtract 2) from 1):
40ah - 400a = 2
a(40h-200) = 2
a = 2/(40h-200)

Next, subtract 3 from 2:
40ah -1200a = 3
a = 3/(40h-1200)

Now, divide the two results for a:
3*(40h-200) = 2*(40h-1200)
120h-600 = 80h-2400
40h = 600-2400
h = -1800/40

Now that you have h, you can easily solve for a using either of the two equations for a, and then substitute to get K.