To complete the input-output table for the function \( f(x) = - (x + 1)^2 + 4 \), we can start by evaluating the function for a set of \( x \) values. Let's choose some values around the vertex of this parabola.
Function Evaluations
The vertex form of a parabola \( f(x) = - a(x - h)^2 + k \) indicates that the output will be largest at \( x = -h \). In this case, since the function is \( f(x) = - (x + 1)^2 + 4 \), the vertex occurs at \( x = -1 \).
Now, let's calculate the function at several \( x \) values, including those around the vertex.
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When \( x = -3 \): \[ f(-3) = -(-3 + 1)^2 + 4 = -(-2)^2 + 4 = -4 + 4 = 0 \]
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When \( x = -2 \): \[ f(-2) = -(-2 + 1)^2 + 4 = -(-1)^2 + 4 = -1 + 4 = 3 \]
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When \( x = -1 \) (the vertex): \[ f(-1) = -(-1 + 1)^2 + 4 = -0^2 + 4 = 0 + 4 = 4 \]
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When \( x = 0 \): \[ f(0) = -(0 + 1)^2 + 4 = -1^2 + 4 = -1 + 4 = 3 \]
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When \( x = 1 \): \[ f(1) = -(1 + 1)^2 + 4 = -2^2 + 4 = -4 + 4 = 0 \]
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When \( x = 2 \): \[ f(2) = -(2 + 1)^2 + 4 = -3^2 + 4 = -9 + 4 = -5 \]
Input-Output Table
Now, we can summarize these calculations in an input-output table:
| \( x \) | \( f(x) \) | |---------|-------------| | -3 | 0 | | -2 | 3 | | -1 | 4 | | 0 | 3 | | 1 | 0 | | 2 | -5 |
Conclusion
From the table, we see that the largest output \( f(x) \) is 4, which occurs when \( x = -1 \). Therefore, \( x = -1 \) produces the largest output for the function \( f(x) = - (x + 1)^2 + 4 \).