To find the maximum or minimum value of the given quadratic function, we first need to rewrite the function in the completed square form. The function is:
y = 2x² + 4x + 7
1. **Factor out the coefficient of x²** from the first two terms:
y = 2(x² + 2x) + 7
2. **Complete the square** for the expression in parentheses. To complete the square for x² + 2x, we take half of the coefficient of x (which is 2), square it (getting 1), and add and subtract this value inside the parentheses:
y = 2(x² + 2x + 1 – 1) + 7
y = 2((x + 1)² – 1) + 7
3. **Distribute the 2** and simplify:
y = 2(x + 1)² – 2 + 7
y = 2(x + 1)² + 5
Now we have the function in the form:
y = 2(x + 1)² + 5
4. **Identify the vertex** of the parabola. Since this is in the form y = a(x – h)² + k, where (h, k) is the vertex, we can see that:
- h = -1
- k = 5
The vertex is at the point (-1, 5).
5. **Determine whether this point is a maximum or minimum**. Since the coefficient ‘a’ (which is 2) is positive, the parabola opens upwards, indicating that the point at (-1, 5) is a minimum point.
6. **State the maximum or minimum value**. Therefore, the minimum value of the quadratic function is 5, which occurs at x = -1.