Question
Describe and correct the error in writing an equation for the function represented by the ordered pairs
(-1, 4),(0, 0),(1, 4),(2, 16),(3, 36)
(-1, 4),(0, 0),(1, 4),(2, 16),(3, 36)
Answers
GPT-4o mini
To find a function that matches the ordered pairs \((-1, 4)\), \((0, 0)\), \((1, 4)\), \((2, 16)\), and \((3, 36)\), we first observe the \(y\)-values relative to the \(x\)-values.
Let's analyze the \(x\) and \(y\) pairs:
- For \(x = -1\), \(y = 4\)
- For \(x = 0\), \(y = 0\)
- For \(x = 1\), \(y = 4\)
- For \(x = 2\), \(y = 16\)
- For \(x = 3\), \(y = 36\)
Looking at the \(y\) values, we notice:
- \(0\) at \(x=0\)
- \(4\) at both \(x=-1\) and \(x=1\)
- \(16\) at \(x=2\) (which is \(4^2\))
- \(36\) at \(x=3\) (which is \(6^2\))
This sequence hints that the function could be related to the square function. Checking the relationship:
- At \(x = 0\): \(0^2 = 0\)
- At \(x = 1\): \(2^2 = 4\)
- At \(x = 2\): \(4^2 = 16\)
- At \(x = 3\): \(6^2 = 36\)
The \(y\)-values appear to correspond to \((2x)^2\) or \(4x^2\) for positive \(x\) and for negative values of \(x\) we see the same behavior.
If we analyze the best fitting function from this, we can hypothesize that the relationship could potentially be:
\[
y = 4x^2
\]
Now, we can test this function against the provided points:
1. For \(x = -1\):
\[
y = 4(-1)^2 = 4(1) = 4 \quad \text{(Matches)}
\]
2. For \(x = 0\):
\[
y = 4(0)^2 = 0 \quad \text{(Matches)}
\]
3. For \(x = 1\):
\[
y = 4(1)^2 = 4(1) = 4 \quad \text{(Matches)}
\]
4. For \(x = 2\):
\[
y = 4(2)^2 = 4(4) = 16 \quad \text{(Matches)}
\]
5. For \(x = 3\):
\[
y = 4(3)^2 = 4(9) = 36 \quad \text{(Matches)}
\]
Thus, the equation representing the function that matches all the ordered pairs is:
\[
\boxed{y = 4x^2}
\]
The error in writing the equation before was likely an oversight in recognizing the quadratic relationship in these values. The correct approach utilizes the formula for a quadratic function, where the coefficients and variable manipulations align properly with the data points provided.
Let's analyze the \(x\) and \(y\) pairs:
- For \(x = -1\), \(y = 4\)
- For \(x = 0\), \(y = 0\)
- For \(x = 1\), \(y = 4\)
- For \(x = 2\), \(y = 16\)
- For \(x = 3\), \(y = 36\)
Looking at the \(y\) values, we notice:
- \(0\) at \(x=0\)
- \(4\) at both \(x=-1\) and \(x=1\)
- \(16\) at \(x=2\) (which is \(4^2\))
- \(36\) at \(x=3\) (which is \(6^2\))
This sequence hints that the function could be related to the square function. Checking the relationship:
- At \(x = 0\): \(0^2 = 0\)
- At \(x = 1\): \(2^2 = 4\)
- At \(x = 2\): \(4^2 = 16\)
- At \(x = 3\): \(6^2 = 36\)
The \(y\)-values appear to correspond to \((2x)^2\) or \(4x^2\) for positive \(x\) and for negative values of \(x\) we see the same behavior.
If we analyze the best fitting function from this, we can hypothesize that the relationship could potentially be:
\[
y = 4x^2
\]
Now, we can test this function against the provided points:
1. For \(x = -1\):
\[
y = 4(-1)^2 = 4(1) = 4 \quad \text{(Matches)}
\]
2. For \(x = 0\):
\[
y = 4(0)^2 = 0 \quad \text{(Matches)}
\]
3. For \(x = 1\):
\[
y = 4(1)^2 = 4(1) = 4 \quad \text{(Matches)}
\]
4. For \(x = 2\):
\[
y = 4(2)^2 = 4(4) = 16 \quad \text{(Matches)}
\]
5. For \(x = 3\):
\[
y = 4(3)^2 = 4(9) = 36 \quad \text{(Matches)}
\]
Thus, the equation representing the function that matches all the ordered pairs is:
\[
\boxed{y = 4x^2}
\]
The error in writing the equation before was likely an oversight in recognizing the quadratic relationship in these values. The correct approach utilizes the formula for a quadratic function, where the coefficients and variable manipulations align properly with the data points provided.