To create an equation from the three points \((0,10)\), \((1,10)\), and \((2,0)\), we can start by identifying the type of function that fits these points. Since we have three points, a quadratic function of the form \(y = ax^2 + bx + c\) could fit the data.
Let's denote the points as:
- \((0, 10)\)
- \((1, 10)\)
- \((2, 0)\)
We can create a system of equations using these points.
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From the point \((0, 10)\): \[ c = 10 \]
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From the point \((1, 10)\): \[ a(1)^2 + b(1) + c = 10 \implies a + b + 10 = 10 \implies a + b = 0 \quad \text{(Equation 1)} \]
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From the point \((2, 0)\): \[ a(2)^2 + b(2) + c = 0 \implies 4a + 2b + 10 = 0 \implies 4a + 2b = -10 \quad \text{(Equation 2)} \]
Now we solve the system of equations.
From Equation 1: \[ b = -a \]
Substituting \(b = -a\) into Equation 2: \[ 4a + 2(-a) = -10 \implies 4a - 2a = -10 \implies 2a = -10 \implies a = -5 \]
Now substituting \(a = -5\) back into Equation 1 to find \(b\): \[ -5 + b = 0 \implies b = 5 \]
Now we have \(a\), \(b\), and \(c\): \[ a = -5, \quad b = 5, \quad c = 10 \]
Thus, the quadratic equation that fits the points \((0,10)\), \((1,10)\), and \((2,0)\) is: \[ y = -5x^2 + 5x + 10 \]
This is the equation based on the given points.
3 answers