To determine the characteristics of the function \( f(x) = x^2 + 2x + 3 \), we can analyze it step by step.
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Identifying the function type:
- The function is a quadratic function in the standard form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 2 \), and \( c = 3 \).
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Finding the vertex:
- The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( a = 1 \) and \( b = 2 \), so: \[ x = -\frac{2}{2 \cdot 1} = -1 \]
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Calculating the minimum value:
- We substitute \( x = -1 \) back into the function to find the minimum value: \[ f(-1) = (-1)^2 + 2(-1) + 3 = 1 - 2 + 3 = 2 \]
Therefore, the minimum value of the function \( f(x) = x^2 + 2x + 3 \) is 2.
Now, we can compare this with the statement:
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Statement A: "The minimum value for both functions is 2."
- True, since \( f(x) \) has a minimum value of 2 and the other function is explicitly stated (assumed comparison) to have a minimum of 2 as well.
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Statement B: "The minimum value for f(x) = x^2 + 2x + 3 is greater than 2."
- False, since the minimum value is exactly 2.
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Statement C: "The minimum value for f(x) = x^2 + 2x + 3 is less than 2."
- False, as the minimum value is exactly 2.
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Statement D: "f(x) = x^2 + 2x + 3 has a maximum value of 2."
- False, as the function has no maximum (it opens upward).
Given the analysis, the correct statement is:
A: The minimum value for both functions is 2.
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