To find the equation of the quadratic function represented by the given table, we will determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the quadratic equation [tex]\(y = ax^2 + bx + c\)[/tex].
Given the points:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-3 & 3.75 \\
\hline
-2 & 4 \\
\hline
-1 & 3.75 \\
\hline
0 & 3 \\
\hline
1 & 1.75 \\
\hline
\end{tabular}
\][/tex]
We use the general form of a quadratic equation:
[tex]\[ y = ax^2 + bx + c \][/tex]
### Step-by-Step Solution:
1. Formulating the Equations:
Using the known points, we substitute each [tex]\((x, y)\)[/tex] pair into the quadratic equation [tex]\(y = ax^2 + bx + c\)[/tex] to get a system of equations.
For [tex]\(x = -3\)[/tex], [tex]\(y = 3.75\)[/tex]:
[tex]\[ 3.75 = a(-3)^2 + b(-3) + c \][/tex]
[tex]\[ 3.75 = 9a - 3b + c \][/tex]
For [tex]\(x = -2\)[/tex], [tex]\(y = 4\)[/tex]:
[tex]\[ 4 = a(-2)^2 + b(-2) + c \][/tex]
[tex]\[ 4 = 4a - 2b + c \][/tex]
For [tex]\(x = -1\)[/tex], [tex]\(y = 3.75\)[/tex]:
[tex]\[ 3.75 = a(-1)^2 + b(-1) + c \][/tex]
[tex]\[ 3.75 = a - b + c \][/tex]
2. Solving the System of Equations:
We solve the simultaneous equations:
[tex]\[
\begin{cases}
9a - 3b + c = 3.75 \\
4a - 2b + c = 4 \\
a - b + c = 3.75 \\
\end{cases}
\][/tex]
The solutions to the system are:
[tex]\[ a = -0.25, \quad b = -1.00, \quad c = 3.00 \][/tex]
Thus, the quadratic equation is:
[tex]\[ y = -0.25x^2 - 1.00x + 3.00 \][/tex]
### Verification:
We can substitute these values back into the original points to verify correctness:
- For [tex]\(x = -3\)[/tex]:
[tex]\[ y = -0.25(-3)^2 - 1(-3) + 3 = -2.25 + 3 + 3 = 3.75 \][/tex]
This matches the given value.
- For [tex]\(x = -2\)[/tex]:
[tex]\[ y = -0.25(-2)^2 - 1(-2) + 3 = -1 + 2 + 3 = 4 \][/tex]
This matches the given value.
- For [tex]\(x = -1\)[/tex]:
[tex]\[ y = -0.25(-1)^2 - 1(-1) + 3 = -0.25 + 1 + 3 = 3.75 \][/tex]
This matches the given value.
Thus, the equation representing the quadratic function is:
[tex]\[ y = -0.25x^2 - 1.00x + 3.00 \][/tex]
So from each drop-down menu, the correct answers are:
[tex]\[ y = -0.25x^2 - 1x + 3 \][/tex]
