Absolutely! To find a function rule that matches the given table of values, we start by noting the pairs of \((x, y)\) values:
[tex]\[
\begin{align*}
(3, 13), \\
(5, 31), \\
(7, 57), \\
(9, 91).
\end{align*}
\][/tex]
We aim to find a polynomial function of the form:
[tex]\[
y = ax^2 + bx + c
\][/tex]
We will determine the coefficients \(a\), \(b\), and \(c\).
### Step-by-Step Solution
1. Set up a system of equations:
For each pair \((x, y)\), substitute the values into the polynomial equation:
[tex]\[
\begin{align*}
13 &= a(3)^2 + b(3) + c, \\
31 &= a(5)^2 + b(5) + c, \\
57 &= a(7)^2 + b(7) + c, \\
91 &= a(9)^2 + b(9) + c.
\end{align*}
\][/tex]
2. Expand and simplify:
[tex]\[
\begin{align*}
13 &= 9a + 3b + c, \\
31 &= 25a + 5b + c, \\
57 &= 49a + 7b + c, \\
91 &= 81a + 9b + c.
\end{align*}
\][/tex]
3. Solve the system of equations:
Solving the above system, we find:
[tex]\[
a = 1, \quad b = 1, \quad c = 1
\][/tex]
### Final Function Rule
From the coefficients obtained, the polynomial function that matches the given table of values is:
[tex]\[
y = x^2 + x + 1
\][/tex]
### Verification
Let's verify by substituting each value of \(x\) back into the function:
1. For \(x = 3\):
[tex]\[
y = 3^2 + 3 + 1 = 9 + 3 + 1 = 13
\][/tex]
2. For \(x = 5\):
[tex]\[
y = 5^2 + 5 + 1 = 25 + 5 + 1 = 31
\][/tex]
3. For \(x = 7\):
[tex]\[
y = 7^2 + 7 + 1 = 49 + 7 + 1 = 57
\][/tex]
4. For \(x = 9\):
[tex]\[
y = 9^2 + 9 + 1 = 81 + 9 + 1 = 91
\][/tex]
Therefore, the function rule [tex]\(y = x^2 + x + 1\)[/tex] correctly matches the given table of values.
