To determine which quadratic equation fits the data presented in the table, we will test each given equation against the provided data points.
Here are the given equations:
1. [tex]\( y = x^2 + x + 3 \)[/tex]
2. [tex]\( y = -x^2 - x - 3 \)[/tex]
3. [tex]\( y = x^2 - x + 3 \)[/tex]
4. [tex]\( y = x^2 - x - 3 \)[/tex]
And the data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-5 & 33 \\
\hline
-2 & 9 \\
\hline
-1 & 5 \\
\hline
0 & 3 \\
\hline
3 & 9 \\
\hline
4 & 15 \\
\hline
6 & 33 \\
\hline
\end{array}
\][/tex]
We will evaluate each equation step by step:
### 1. Testing [tex]\( y = x^2 + x + 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = (-5)^2 + (-5) + 3 = 25 - 5 + 3 = 23 \quad (\text{Given: } 33)
\][/tex]
Since 23 ≠ 33, this equation does not fit.
### 2. Testing [tex]\( y = -x^2 - x - 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = -(-5)^2 - (-5) - 3 = -25 + 5 - 3 = -23 \quad (\text{Given: } 33)
\][/tex]
Since -23 ≠ 33, this equation does not fit.
### 3. Testing [tex]\( y = x^2 - x + 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = (-5)^2 - (-5) + 3 = 25 + 5 + 3 = 33 \quad (\text{Given: } 33)
\][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9 \quad (\text{Given: } 9)
\][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = (-1)^2 - (-1) + 3 = 1 + 1 + 3 = 5 \quad (\text{Given: } 5)
\][/tex]
For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = (0)^2 - 0 + 3 = 0 - 0 + 3 = 3 \quad (\text{Given: } 3)
\][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = (3)^2 - 3 + 3 = 9 - 3 + 3 = 9 \quad (\text{Given: } 9)
\][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = (4)^2 - 4 + 3 = 16 - 4 + 3 = 15 \quad (\text: } 15)
\][/tex]
For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = (6)^2 - 6 + 3 = 36 - 6 + 3 = 33 \quad (\text{Given: } 33)
\][/tex]
Since all the calculations match the provided data points, this equation fits.
### 4. Testing [tex]\( y = x^2 - x - 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = (-5)^2 - (-5) - 3 = 25 + 5 - 3 = 27 \quad (\text{Given: } 33)
\][/tex]
Since 27 ≠ 33, this equation does not fit.
Thus, the quadratic equation that fits the given data is:
[tex]\[
y = x^2 - x + 3
\][/tex]
So the correct answer is:
[tex]\( y = x^2 - x + 3 \)[/tex] which corresponds to option 3.
