To find the coefficients \(a\) and \(b\) for the quadratic function \(f(x) = ax^2 + bx + c\) where \(c = 0\), we can set up the system of equations based on the given points on the graph.
You mention that the area of the rectangle (which can be represented as \(f(x)\)) corresponding to a given width represented by \(x\) can be calculated as follows:
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We have the point \((3, 15)\), which means when \(x = 3\), \(f(3) = 15\): \[ a(3^2) + b(3) = 15 \] This simplifies to: \[ 9a + 3b = 15 \quad \text{(Equation 1)} \]
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The other point given is \((6, 12)\), so when \(x = 6\), \(f(6) = 12\): \[ a(6^2) + b(6) = 12 \] This simplifies to: \[ 36a + 6b = 12 \quad \text{(Equation 2)} \]
Now we have the following system of linear equations:
- \(9a + 3b = 15\)
- \(36a + 6b = 12\)
Thus, the correct system of linear equations that can be used to identify the coefficients \(a\) and \(b\) is: \[ 9a + 3b = 15 \quad \text{and} \quad 36a + 6b = 12 \]
So the answer is: 9a + 3b = 15 and 36a + 6b = 12.