Determine the vertex form and the maximum or minimum value of the function. \[ f(x)=2 x^{2}+8 x+3 \] A \( f(x)=2(x+4)^{2}-13 \), the maximum value of \( f \) is -13 B \( f(x)=2(x+4)^{2}-13 \), the minimum value of \( f \) is -13 C \( f(x)=2(x+2)^{2}-5 \), the maximum value of \( f \) is -5 (D) \( f(x)=2(x+2)^{2}-5 \), the minimum value of \( f \) is -5

To determine the **vertex form** of the function \( f(x) = 2x^{2} + 8x + 3 \) and identify whether it has a **maximum or minimum** value, follow these steps: ### 1. **Convert to Vertex Form** The vertex form of a quadratic function is: \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. **Given Function:** \[ f(x) = 2x^{2} + 8x + 3 \] **Step-by-Step Conversion:** 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ f(x) = 2(x^{2} + 4x) + 3 \] 2. **Complete the square inside the parentheses:** - Take half of the coefficient of \(x\): \(\frac{4}{2} = 2\) - Square it: \(2^2 = 4\) Add and subtract this square inside the parentheses: \[ f(x) = 2(x^{2} + 4x + 4 - 4) + 3 \] \[ f(x) = 2((x + 2)^2 - 4) + 3 \] 3. **Simplify the expression:** \[ f(x) = 2(x + 2)^2 - 8 + 3 \] \[ f(x) = 2(x + 2)^2 - 5 \] ### 2. **Identify Vertex and Determine Maximum or Minimum** - **Vertex:** \((-2, -5)\) - **Coefficient of \((x + 2)^2\):** \(2\) (positive) Since the coefficient is positive, the parabola **opens upwards**, indicating that the function has a **minimum** value. - **Minimum Value:** \(-5\) ### 3. **Conclusion** The correct vertex form and the nature of the extremum (minimum value) is: **(D)** \[ f(x) = 2(x + 2)^{2} - 5 \] **The minimum value of \( f \) is -5.** Answer: **(D) \( f(x)=2(x+2)^{2}-5 \), the minimum value of \( f \) is -5**