You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, [tex]\( y \)[/tex], after [tex]\( x \)[/tex] seconds.

[tex]\[
\begin{array}{|c|c|}
\hline
x \text{ (seconds after the ball is thrown)} & y \text{ (ball's height in feet above the ground)} \\
\hline
1 & 359 \\
\hline
3 & 281 \\
\hline
4 & 194 \\
\hline
\end{array}
\][/tex]

a. Find the quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] whose graph passes through the given points.

[tex]\( y = \boxed{\quad} \)[/tex]



Answer :

To find the quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] whose graph passes through the given points [tex]\((1, 359)\)[/tex], [tex]\((3, 281)\)[/tex], and [tex]\((4, 194)\)[/tex], we need to determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. Let's consider each given point and substitute them into the quadratic equation:

For the point [tex]\((1, 359)\)[/tex]:
[tex]\[ 359 = a(1^2) + b(1) + c \][/tex]
Which simplifies to:
[tex]\[ a + b + c = 359 \quad \text{(Equation 1)} \][/tex]

For the point [tex]\((3, 281)\)[/tex]:
[tex]\[ 281 = a(3^2) + b(3) + c \][/tex]
Which simplifies to:
[tex]\[ 9a + 3b + c = 281 \quad \text{(Equation 2)} \][/tex]

For the point [tex]\((4, 194)\)[/tex]:
[tex]\[ 194 = a(4^2) + b(4) + c \][/tex]
Which simplifies to:
[tex]\[ 16a + 4b + c = 194 \quad \text{(Equation 3)} \][/tex]

Now, we can solve this system of equations:

[tex]\[ \begin{cases} a + b + c = 359 \\ 9a + 3b + c = 281 \\ 16a + 4b + c = 194 \\ \end{cases} \][/tex]

Step 1: Subtract Equation 1 from Equation 2 to eliminate [tex]\( c \)[/tex]:
[tex]\[ (9a + 3b + c) - (a + b + c) = 281 - 359 \\ 8a + 2b = -78 \\ 4a + b = -39 \quad \text{(Equation 4)} \][/tex]

Step 2: Subtract Equation 2 from Equation 3 to eliminate [tex]\( c \)[/tex]:
[tex]\[ (16a + 4b + c) - (9a + 3b + c) = 194 - 281 \\ 7a + b = -87 \quad \text{(Equation 5)} \][/tex]

Step 3: Subtract Equation 4 from Equation 5 to eliminate [tex]\( b \)[/tex]:
[tex]\[ (7a + b) - (4a + b) = -87 - (-39) \\ 3a = -48 \\ a = -16 \][/tex]

Step 4: Substitute [tex]\( a = -16 \)[/tex] back into Equation 4 to find [tex]\( b \)[/tex]:
[tex]\[ 4(-16) + b = -39 \\ -64 + b = -39 \\ b = 25 \][/tex]

Step 5: Substitute [tex]\( a = -16 \)[/tex] and [tex]\( b = 25 \)[/tex] back into Equation 1 to find [tex]\( c \)[/tex]:
[tex]\[ -16 + 25 + c = 359 \\ 9 + c = 359 \\ c = 350 \][/tex]

Thus, the quadratic function is:
[tex]\[ y = -16x^2 + 25x + 350 \][/tex]