Question
Create an equation that models the following dataset. Explain why the equation correctly models the dataset.
x −3
−2
−1
0 1 2 3
y 18 8 2 0 2 8 18
x −3
−2
−1
0 1 2 3
y 18 8 2 0 2 8 18
Answers
GPT-4o mini
To model the provided dataset, we can observe the values of \( x \) and their corresponding \( y \) values:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 18 \\
-2 & 8 \\
-1 & 2 \\
0 & 0 \\
1 & 2 \\
2 & 8 \\
3 & 18 \\
\hline
\end{array}
\]
### Pattern Analysis
Let's analyze the dataset:
1. The \( y \) values appear to be symmetric about \( x = 0 \). Specifically, notice that:
- \( y(-3) = y(3) = 18 \)
- \( y(-2) = y(2) = 8 \)
- \( y(-1) = y(1) = 2 \)
- \( y(0) = 0 \)
2. This symmetry suggests that the function might be a quadratic function, which generally has the form:
\[
y = ax^2 + bx + c
\]
However, since it is symmetric about the y-axis (even function), the term \( bx \) will be zero, reducing the equation to:
\[
y = ax^2 + c
\]
3. From the data, we can see that when \( x = 0 \), \( y = 0 \). Therefore, \( c = 0 \) and our equation simplifies further to:
\[
y = ax^2
\]
### Finding the Coefficient \( a \)
To find the coefficient \( a \), we can use the points we have. We can use any point from the dataset; let's use the point \( (1, 2) \):
\[
2 = a(1^2)
\]
\[
2 = a
\]
Thus, we find that \( a = 2 \). Therefore, the equation is:
\[
y = 2x^2
\]
### Verification with Dataset
We can verify that this equation correctly models all the points in the dataset:
1. For \( x = -3 \):
\[
y = 2(-3)^2 = 2(9) = 18
\]
2. For \( x = -2 \):
\[
y = 2(-2)^2 = 2(4) = 8
\]
3. For \( x = -1 \):
\[
y = 2(-1)^2 = 2(1) = 2
\]
4. For \( x = 0 \):
\[
y = 2(0)^2 = 0
\]
5. For \( x = 1 \):
\[
y = 2(1)^2 = 2(1) = 2
\]
6. For \( x = 2 \):
\[
y = 2(2)^2 = 2(4) = 8
\]
7. For \( x = 3 \):
\[
y = 2(3)^2 = 2(9) = 18
\]
### Conclusion
Thus, the equation that models the dataset is:
\[
y = 2x^2
\]
This equation correctly models the dataset because it matches all the \( y \) values for corresponding \( x \) values and exhibits the symmetry observed in the dataset. Hence, this quadratic function accurately represents the relationship between \( x \) and \( y \) in the provided dataset.
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 18 \\
-2 & 8 \\
-1 & 2 \\
0 & 0 \\
1 & 2 \\
2 & 8 \\
3 & 18 \\
\hline
\end{array}
\]
### Pattern Analysis
Let's analyze the dataset:
1. The \( y \) values appear to be symmetric about \( x = 0 \). Specifically, notice that:
- \( y(-3) = y(3) = 18 \)
- \( y(-2) = y(2) = 8 \)
- \( y(-1) = y(1) = 2 \)
- \( y(0) = 0 \)
2. This symmetry suggests that the function might be a quadratic function, which generally has the form:
\[
y = ax^2 + bx + c
\]
However, since it is symmetric about the y-axis (even function), the term \( bx \) will be zero, reducing the equation to:
\[
y = ax^2 + c
\]
3. From the data, we can see that when \( x = 0 \), \( y = 0 \). Therefore, \( c = 0 \) and our equation simplifies further to:
\[
y = ax^2
\]
### Finding the Coefficient \( a \)
To find the coefficient \( a \), we can use the points we have. We can use any point from the dataset; let's use the point \( (1, 2) \):
\[
2 = a(1^2)
\]
\[
2 = a
\]
Thus, we find that \( a = 2 \). Therefore, the equation is:
\[
y = 2x^2
\]
### Verification with Dataset
We can verify that this equation correctly models all the points in the dataset:
1. For \( x = -3 \):
\[
y = 2(-3)^2 = 2(9) = 18
\]
2. For \( x = -2 \):
\[
y = 2(-2)^2 = 2(4) = 8
\]
3. For \( x = -1 \):
\[
y = 2(-1)^2 = 2(1) = 2
\]
4. For \( x = 0 \):
\[
y = 2(0)^2 = 0
\]
5. For \( x = 1 \):
\[
y = 2(1)^2 = 2(1) = 2
\]
6. For \( x = 2 \):
\[
y = 2(2)^2 = 2(4) = 8
\]
7. For \( x = 3 \):
\[
y = 2(3)^2 = 2(9) = 18
\]
### Conclusion
Thus, the equation that models the dataset is:
\[
y = 2x^2
\]
This equation correctly models the dataset because it matches all the \( y \) values for corresponding \( x \) values and exhibits the symmetry observed in the dataset. Hence, this quadratic function accurately represents the relationship between \( x \) and \( y \) in the provided dataset.