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]
