To create a quadratic model for the dataset in the xy-chart, we need to find a quadratic function of the form \( f(x) = ax^2 + bx + c \) that fits the given data points.
The provided points are:
- \( (-1, 6) \)
- \( (0, 5) \)
- \( (1, 6) \)
- \( (2, 9) \)
- \( (3, 14) \)
Step 1: Identify Parameters
We can substitute the known points into the quadratic equation to find \( a, b, \) and \( c \).
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Using point (0, 5): \[ f(0) = a(0)^2 + b(0) + c = c \implies c = 5 \]
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Now use point (1, 6): \[ f(1) = a(1)^2 + b(1) + c = 6 \implies a + b + 5 = 6 \implies a + b = 1 \quad (1) \]
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Using point (2, 9): \[ f(2) = a(2)^2 + b(2) + c = 9 \implies 4a + 2b + 5 = 9 \implies 4a + 2b = 4 \implies 2a + b = 2 \quad (2) \]
Step 2: Solve the System of Equations
Now we have a system of two equations from steps 2 and 3:
- \( a + b = 1 \) (Equation 1)
- \( 2a + b = 2 \) (Equation 2)
We can subtract Equation 1 from Equation 2: \[ (2a + b) - (a + b) = 2 - 1 \implies a = 1 \]
Now we can substitute \( a = 1 \) back into Equation 1: \[ 1 + b = 1 \implies b = 0 \]
Step 3: Construct the Function
Now we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = 5 \)
Thus, the quadratic model is: \[ f(x) = 1x^2 + 0x + 5 = x^2 + 5 \]
Conclusion
The quadratic model for the dataset is:
f(x) = x² + 5
This corresponds to the response: f(x) = x² + 5