To find the quadratic function that best fits the given data, we will assume the function has the form:
\[ f(x) = ax^2 + bx + c \]
We have the data points:
\[ \begin{align*} (0, 0) \ (2, 397) \ (4, 1602) \ (6, 3602) \ (8, 6402) \ (10, 9999) \ \end{align*} \]
We'll create a system of equations based on these points to solve for \(a\), \(b\), and \(c\).
Step 1: Set Up the System of Equations
Using the points, we can substitute into the equation \(f(x) = ax^2 + bx + c\):
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For \(x = 0\), \(f(0) = 0\): \[ c = 0 \]
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For \(x = 2\), \(f(2) = 397\): \[ 4a + 2b + c = 397 \ 4a + 2b + 0 = 397 \quad (c = 0) \ 4a + 2b = 397 \quad \text{(Eq. 1)} \]
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For \(x = 4\), \(f(4) = 1602\): \[ 16a + 4b + c = 1602 \ 16a + 4b + 0 = 1602 \quad (c = 0) \ 16a + 4b = 1602 \quad \text{(Eq. 2)} \]
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For \(x = 6\), \(f(6) = 3602\): \[ 36a + 6b + c = 3602 \ 36a + 6b + 0 = 3602 \quad (c = 0) \ 36a + 6b = 3602 \quad \text{(Eq. 3)} \]
Step 2: Solve the System
First, we can eliminate \(b\) by solving Eq. 1 and Eq. 2:
From Eq. 1: \[ 2b = 397 - 4a \ b = \frac{397 - 4a}{2} \]
Substitute \(b\) into Eq. 2: \[ 16a + 4\left(\frac{397 - 4a}{2}\right) = 1602 \ 16a + 2(397 - 4a) = 1602 \ 16a + 794 - 8a = 1602 \ 8a = 1602 - 794 \ 8a = 808 \ a = 101 \]
Now substitute \(a\) back into Eq. 1 to find \(b\): \[ 4(101) + 2b = 397 \ 404 + 2b = 397 \ 2b = 397 - 404 \ 2b = -7 \ b = -3.5 \]
Summary of the Coefficients
- \(a = 101\)
- \(b = -3.5\)
- \(c = 0\)
Step 3: Write the Quadratic Function
Now we can write the quadratic function: \[ f(x) = 101x^2 - 3.5x + 0 \]
So, the best-fit quadratic function is:
\[ f(x) = 101x^2 - 3.5x \]
Final Answer Format
The quadratic function in the requested format is:
\[ y = 101x^2 - 3.5x + 0 \]
The values to fill in are:
- \(a = 101\)
- \(b = -3.5\)
- \(c = 0\)
Write it as:
\[ y = 101x^2 - 3.5x + 0 \]