Question
Create an equation that models the following dataset. Explain why the equation correctly models the dataset.
The table displays the pairs of input-output for symmetric function.
x −3, -2, -1, 0, 1, 2, 3
y 18, 8, 2, 0, 2, 8, 18
The table displays the pairs of input-output for symmetric function.
x −3, -2, -1, 0, 1, 2, 3
y 18, 8, 2, 0, 2, 8, 18
Answers
GPT-4o mini
To model the given dataset, we first need to analyze the symmetry in the output values. The values provided are:
\[
\begin{align*}
x: & -3, -2, -1, 0, 1, 2, 3 \\
y: & 18, 8, 2, 0, 2, 8, 18 \\
\end{align*}
\]
From this, we can observe that the function exhibits symmetry around \(x = 0\). This suggests that the relationship might be represented by an even function (a function where \(f(-x) = f(x)\)), and potentially a quadratic function since quadratics are the simplest forms of even functions.
Next, we can look for a quadratic function of the form:
\[
y = ax^2 + bx + c
\]
However, since the function is symmetric about the y-axis, we can simplify it to:
\[
y = ax^2 + c
\]
Now we need to determine the coefficients \(a\) and \(c\). We can use the known points in the dataset to find these coefficients:
1. At \(x = 0\):
\[
y = c = 0 \quad \implies c = 0
\]
Thus, our equation becomes:
\[
y = ax^2
\]
2. We can use another point to find \(a\). Let's use the point \((1, 2)\):
\[
2 = a(1)^2 \implies a = 2
\]
So now we have the equation:
\[
y = 2x^2
\]
Now, we can verify that this equation models other points in the dataset correctly:
- For \(x = -3\):
\[
y = 2(-3)^2 = 2 \times 9 = 18
\]
- For \(x = -2\):
\[
y = 2(-2)^2 = 2 \times 4 = 8
\]
- For \(x = -1\):
\[
y = 2(-1)^2 = 2 \times 1 = 2
\]
- For \(x = 0\):
\[
y = 2(0)^2 = 0
\]
- For \(x = 1\):
\[
y = 2(1)^2 = 2 \times 1 = 2
\]
- For \(x = 2\):
\[
y = 2(2)^2 = 2 \times 4 = 8
\]
- For \(x = 3\):
\[
y = 2(3)^2 = 2 \times 9 = 18
\]
All the calculated values match the \(y\) values from the dataset.
### Conclusion:
Thus, the equation that models the dataset is
\[
y = 2x^2
\]
This equation correctly reflects the pairs in the dataset due to its symmetric nature about the y-axis and reproduces all corresponding output values for each input.
\[
\begin{align*}
x: & -3, -2, -1, 0, 1, 2, 3 \\
y: & 18, 8, 2, 0, 2, 8, 18 \\
\end{align*}
\]
From this, we can observe that the function exhibits symmetry around \(x = 0\). This suggests that the relationship might be represented by an even function (a function where \(f(-x) = f(x)\)), and potentially a quadratic function since quadratics are the simplest forms of even functions.
Next, we can look for a quadratic function of the form:
\[
y = ax^2 + bx + c
\]
However, since the function is symmetric about the y-axis, we can simplify it to:
\[
y = ax^2 + c
\]
Now we need to determine the coefficients \(a\) and \(c\). We can use the known points in the dataset to find these coefficients:
1. At \(x = 0\):
\[
y = c = 0 \quad \implies c = 0
\]
Thus, our equation becomes:
\[
y = ax^2
\]
2. We can use another point to find \(a\). Let's use the point \((1, 2)\):
\[
2 = a(1)^2 \implies a = 2
\]
So now we have the equation:
\[
y = 2x^2
\]
Now, we can verify that this equation models other points in the dataset correctly:
- For \(x = -3\):
\[
y = 2(-3)^2 = 2 \times 9 = 18
\]
- For \(x = -2\):
\[
y = 2(-2)^2 = 2 \times 4 = 8
\]
- For \(x = -1\):
\[
y = 2(-1)^2 = 2 \times 1 = 2
\]
- For \(x = 0\):
\[
y = 2(0)^2 = 0
\]
- For \(x = 1\):
\[
y = 2(1)^2 = 2 \times 1 = 2
\]
- For \(x = 2\):
\[
y = 2(2)^2 = 2 \times 4 = 8
\]
- For \(x = 3\):
\[
y = 2(3)^2 = 2 \times 9 = 18
\]
All the calculated values match the \(y\) values from the dataset.
### Conclusion:
Thus, the equation that models the dataset is
\[
y = 2x^2
\]
This equation correctly reflects the pairs in the dataset due to its symmetric nature about the y-axis and reproduces all corresponding output values for each input.