To determine which quadratic equation fits the given data points, we need to evaluate each of the provided quadratic equations at the given [tex]\( x \)[/tex] values and check if the resulting [tex]\( y \)[/tex] is as specified in the table. The data points given are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & -11 \\
\hline
-2 & -9 \\
\hline
-1 & -5 \\
\hline
0 & 1 \\
\hline
1 & 9 \\
\hline
3 & 31 \\
\hline
6 & 79 \\
\hline
\end{array}
\][/tex]
We will consider each quadratic equation in turn.
### Equation 1: [tex]\( y = x^2 - 7x + 1 \)[/tex]
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = (-3)^2 - 7(-3) + 1 = 9 + 21 + 1 = 31 \quad (\text{not } -11)
\][/tex]
This equation does not fit the data as the [tex]\( y \)[/tex] value for [tex]\( x = -3 \)[/tex] does not match.
### Equation 2: [tex]\( y = x^2 - 7x - 1 \)[/tex]
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = (-3)^2 - 7(-3) - 1 = 9 + 21 - 1 = 29 \quad (\text{not } -11)
\][/tex]
This equation does not fit the data as the [tex]\( y \)[/tex] value for [tex]\( x = -3 \)[/tex] does not match.
### Equation 3: [tex]\( y = x^2 + 7x + 1 \)[/tex]
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = (-3)^2 + 7(-3) + 1 = 9 - 21 + 1 = -11 \quad (\text{matches})
\][/tex]
2. For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = (-2)^2 + 7(-2) + 1 = 4 - 14 + 1 = -9 \quad (\text{matches})
\][/tex]
3. For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = (-1)^2 + 7(-1) + 1 = 1 - 7 + 1 = -5 \quad (\text{matches})
\][/tex]
4. For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0^2 + 7(0) + 1 = 0 + 0 + 1 = 1 \quad (\text{matches})
\][/tex]
5. For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 1^2 + 7(1) + 1 = 1 + 7 + 1 = 9 \quad (\text{matches})
\][/tex]
6. For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 3^2 + 7(3) + 1 = 9 + 21 + 1 = 31 \quad (\text{matches})
\][/tex]
7. For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = 6^2 + 7(6) + 1 = 36 + 42 + 1 = 79 \quad (\text{matches})
\][/tex]
This equation fits all the data points.
### Equation 4: [tex]\( y = -x^2 + 7x + 1 \)[/tex]
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -(-3)^2 + 7(-3) + 1 = -9 - 21 + 1 = -29 \quad (\text{not } -11)
\][/tex]
This equation does not fit the data as the [tex]\( y \)[/tex] value for [tex]\( x = -3 \)[/tex] does not match.
Conclusion: The quadratic equation that fits all the given data points is:
[tex]\[
y = x^2 + 7x + 1
\][/tex]
