To analyze the given quadratic function \( f(x) = - (x + 1.4)^2 + 6 \), let's go through the parts step-by-step.
Part A: Identify the vertex. The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. The function you've provided is:
\[ f(x) = - (x + 1.4)^2 + 6 \]
From this equation, we can identify:
- The term \( (x + 1.4) \) indicates that \( h = -1.4 \).
- The maximum value of the function occurs when \( x + 1.4 = 0 \), which corresponds to \( k = 6 \).
So, the vertex is:
\[ \text{Vertex} = (-1.4, 6) \]
Part B: Complete the table.
To complete the table, we will calculate the values of \( f(x) \) for \( x = 0, 1, 2, 3 \).
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For \( x = 0 \): \[ f(0) = - (0 + 1.4)^2 + 6 = - (1.4)^2 + 6 = -1.96 + 6 = 4.04 \]
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For \( x = 1 \): \[ f(1) = - (1 + 1.4)^2 + 6 = - (2.4)^2 + 6 = -5.76 + 6 = 0.24 \]
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For \( x = 2 \): \[ f(2) = - (2 + 1.4)^2 + 6 = - (3.4)^2 + 6 = -11.56 + 6 = -5.56 \]
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For \( x = 3 \): \[ f(3) = - (3 + 1.4)^2 + 6 = - (4.4)^2 + 6 = -19.36 + 6 = -13.36 \]
Now we can complete the table:
| \(x\) | \(f(x)\) | |:----:|:------:| | 0 | 4.04 | | 1 | 0.24 | | 2 | -5.56 | | 3 | -13.36 |
Summary:
- Part A: Vertex = \((-1.4, 6)\)
- Part B: | \(x\) | \(f(x)\) | |:----:|:------:| | 0 | 4.04 | | 1 | 0.24 | | 2 | -5.56 | | 3 | -13.36 |
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