Question

a human cannonball is blasted out in a parabolic shape relative to the ground. the table below shows his height (in meters) over a given time (in seconds).
find the quadratic model of the table provided. round to the nearest hundredth.
model: ( f ( x ) = ) type your answer... ( x ^ { 2 } + ) type your answer... ( x + ) type your answer...

Explanation:

Step1: Set up the system of equations

Let the quadratic model be \(f(x)=ax^{2}+bx + c\).
Take three points \((x_1,y_1)=(0.5,67)\), \((x_2,y_2)=(1,74)\), \((x_3,y_3)=(1.5,77)\)
Substitute into the model:
For \(x = 0.5,y=67\): \(67=a(0.5)^{2}+b(0.5)+c\), i.e., \(67 = 0.25a+0.5b + c\)
For \(x = 1,y=74\): \(74=a(1)^{2}+b(1)+c\), i.e., \(74=a + b + c\)
For \(x = 1.5,y=77\): \(77=a(1.5)^{2}+b(1.5)+c\), i.e., \(77 = 2.25a+1.5b + c\)

Step2: Solve the system of equations

Subtract the first equation from the second:
\((a + b + c)-(0.25a+0.5b + c)=74 - 67\)
\(a + b + c-0.25a - 0.5b - c=7\)
\(0.75a+0.5b=7\) (Equation A)
Subtract the second equation from the third:
\((2.25a+1.5b + c)-(a + b + c)=77 - 74\)
\(2.25a+1.5b + c-a - b - c = 3\)
\(1.25a+0.5b=3\) (Equation B)
Subtract Equation B from Equation A:
\((0.75a+0.5b)-(1.25a+0.5b)=7 - 3\)
\(0.75a+0.5b-1.25a - 0.5b=4\)
\(- 0.5a=4\), so \(a=-8\)
Substitute \(a = - 8\) into Equation A:
\(0.75\times(-8)+0.5b=7\)
\(-6 + 0.5b=7\)
\(0.5b=13\), so \(b = 26\)
Substitute \(a=-8,b = 26\) into \(74=a + b + c\)
\(74=-8+26 + c\)
\(c=56\)

Answer:

\(f(x)=-8x^{2}+26x + 56\)