Question

Compare f(x) =
+ 2x + 3 to a function that has a minimum value of 2. Which statement is TRUE?
Responses
A The minimum value for both functions is 2.The minimum value for both functions is 2.
B The minimum value for f(x) =
+ 2x + 3 is greater than 2.The minimum value for f(x) = x 2 + 2x + 3 is greater than 2.
C The minimum value for f(x) =
+ 2x + 3 is less than 2.The minimum value for f(x) = x 2 + 2x + 3 is less than 2.
D f(x) =
+ 2x + 3 has a maximum value of 2.f(x) = x 2 + 2x + 3 has a maximum value of 2.

Answer 1

To determine the characteristics of the function \( f(x) = x^2 + 2x + 3 \), we can analyze it step by step.

1. **Identifying the function type:**
- The function is a quadratic function in the standard form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 2 \), and \( c = 3 \).

2. **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:
\[
x = -\frac{2}{2 \cdot 1} = -1
\]

3. **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:

1. **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.

2. **Statement B:** "The minimum value for f(x) = x^2 + 2x + 3 is greater than 2."
- **False**, since the minimum value is exactly 2.

3. **Statement C:** "The minimum value for f(x) = x^2 + 2x + 3 is less than 2."
- **False**, as the minimum value is exactly 2.

4. **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.**