To find the equation of the quadratic function of the form \( y = ax^2 + bx + c \) that best fits the given points \((-3, -40)\), \((1, 12)\), \((5, 72)\), and \((7, 137)\), we can use the method of least squares.
Step 1: Set up the equations
We have four points, which means we can set up a system of equations based on those points:
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For \((-3, -40)\): \[ -40 = a(-3)^2 + b(-3) + c \] Simplifying, we get: \[ -40 = 9a - 3b + c \quad (1) \]
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For \((1, 12)\): \[ 12 = a(1)^2 + b(1) + c \] Simplifying, we get: \[ 12 = a + b + c \quad (2) \]
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For \((5, 72)\): \[ 72 = a(5)^2 + b(5) + c \] Simplifying, we get: \[ 72 = 25a + 5b + c \quad (3) \]
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For \((7, 137)\): \[ 137 = a(7)^2 + b(7) + c \] Simplifying, we get: \[ 137 = 49a + 7b + c \quad (4) \]
Step 2: Form the system of equations
Now we can write the system of equations in terms of a, b, and c:
- \( 9a - 3b + c = -40 \) (Eq. 1)
- \( a + b + c = 12 \) (Eq. 2)
- \( 25a + 5b + c = 72 \) (Eq. 3)
- \( 49a + 7b + c = 137 \) (Eq. 4)
Step 3: Eliminate variable \(c\)
We can eliminate \(c\) from these equations by subtracting the equations as follows:
Subtract (2) from (1): \[ (9a - 3b + c) - (a + b + c) = -40 - 12 \] This simplifies to: \[ 8a - 4b = -52 \quad (5) \]
Subtract (2) from (3): \[ (25a + 5b + c) - (a + b + c) = 72 - 12 \] This simplifies to: \[ 24a + 4b = 60 \quad (6) \]
Subtract (3) from (4): \[ (49a + 7b + c) - (25a + 5b + c) = 137 - 72 \] This simplifies to: \[ 24a + 2b = 65 \quad (7) \]
Step 4: Solve the resulting system
Now we have a simpler system of equations:
- \( 8a - 4b = -52 \) (5)
- \( 24a + 4b = 60 \) (6)
- \( 24a + 2b = 65 \) (7)
From (5): \[ 2a - b = -13 \quad (8) \]
From (6): \[ 24a + 4b = 60 \quad (6) \implies 6a + b = 15 \quad (9) \] From (7): \[ 24a + 2b = 65 \quad (7) \implies 12a + b = 32.5 \quad (10) \]
Step 5: Solve (8) and (9):
We can solve (8) and (9) simultaneously:
From (8): \[ b = 2a + 13 \quad (11) \]
Substituting (11) into (9): \[ 6a + (2a + 13) = 15 \implies 8a + 13 = 15 \implies 8a = 2 \implies a = \frac{1}{4} \]
Substituting \(a\) back to find \(b\): \[ b = 2\left(\frac{1}{4}\right) + 13 = \frac{1}{2} + 13 = \frac{27}{2} \]
Step 6: Find \(c\)
Now substitute \(a\) and \(b\) back into any of the original equations to find \(c\). Using equation (2): \[ \frac{1}{4} + \frac{27}{2} + c = 12 \] This results in: \[ c = 12 - \frac{1}{4} - \frac{27}{2} \] To perform subtraction, convert everything to a common denominator: \[ = 12 - \frac{1}{4} - \frac{54}{4} = 12 - \frac{55}{4} = \frac{48}{4} - \frac{55}{4} = -\frac{7}{4} \]
Final Equation
Thus, the coefficients are:
- \(a = \frac{1}{4}\)
- \(b = \frac{27}{2}\)
- \(c = -\frac{7}{4}\)
The equation of the line of best fit is: \[ y = \frac{1}{4}x^2 + \frac{27}{2}x - \frac{7}{4} \]
This is the required quadratic function that fits the given points.
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