To determine a quadratic regression for the data points \((1, 5)\), \((3, 4)\), \((4, 1)\), \((5, 2)\), and \((7, 4)\), we will use the general form of a quadratic equation, which is:
\[ y = ax^2 + bx + c \]
Where \(a\), \(b\), and \(c\) are coefficients that we need to find.
Step 1: Set up the system of equations
We start by creating a system of equations based on the given data points:
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For the point \((1, 5)\): \[ 5 = a(1)^2 + b(1) + c \implies a + b + c = 5 \quad \text{(1)} \]
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For the point \((3, 4)\): \[ 4 = a(3)^2 + b(3) + c \implies 9a + 3b + c = 4 \quad \text{(2)} \]
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For the point \((4, 1)\): \[ 1 = a(4)^2 + b(4) + c \implies 16a + 4b + c = 1 \quad \text{(3)} \]
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For the point \((5, 2)\): \[ 2 = a(5)^2 + b(5) + c \implies 25a + 5b + c = 2 \quad \text{(4)} \]
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For the point \((7, 4)\): \[ 4 = a(7)^2 + b(7) + c \implies 49a + 7b + c = 4 \quad \text{(5)} \]
Step 2: Solve the system of equations
We'll start solving with equations (1), (2), and (3):
From (1): \[ c = 5 - a - b \]
Substituting \(c\) into (2): \[ 9a + 3b + (5 - a - b) = 4 \] \[ 8a + 2b + 5 = 4 \implies 8a + 2b = -1 \implies 4a + b = -\frac{1}{2} \quad \text{(6)} \]
Substituting \(c\) into (3): \[ 16a + 4b + (5 - a - b) = 1 \] \[ 15a + 3b + 5 = 1 \implies 15a + 3b = -4 \implies 5a + b = -\frac{4}{3} \quad \text{(7)} \]
Step 3: Eliminate \(b\)
Subtract (6) from (7): \[ (5a + b) - (4a + b) = -\frac{4}{3} + \frac{1}{2} \] \[ a = -\frac{4}{3} + \frac{3}{6} = -\frac{8}{6} + \frac{3}{6} = -\frac{5}{6} \]
Step 4: Find \(b\) and \(c\)
Substituting \(a\) back into (6): \[ 4(-\frac{5}{6}) + b = -\frac{1}{2} \implies -\frac{20}{6} + b = -\frac{1}{2} \] \[ b = -\frac{1}{2} + \frac{10}{3} = -\frac{3}{6} + \frac{20}{6} = \frac{17}{6} \]
Substituting \(a\) and \(b\) back into (1) to find \(c\): \[ -\frac{5}{6} + \frac{17}{6} + c = 5 \] \[ c = 5 - \frac{12}{6} = 5 - 2 = 3 \]
Step 5: Final result
Now we have the coefficients: \[ a = -\frac{5}{6}, \quad b = \frac{17}{6}, \quad c = 3 \]
Thus, the quadratic regression function is:
\[ y = -\frac{5}{6}x^2 + \frac{17}{6}x + 3 \]
This represents the best fitting quadratic curve for the given data points.
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