How can I find the maximum or minimum value of the quadratic function y = 2x² + 4x + 7 by completing the square, and what is the value of x at which the function reaches its maximum or minimum?

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Finding the Maximum or Minimum Value of the Quadratic Function

To find the maximum or minimum value of the quadratic function y = 2x² + 4x + 7, we can use the method of completing the square. Completing the square not only allows us to rewrite the function in a vertex form but also helps us easily identify the maximum or minimum point.

Step 1: Factor Out the Leading Coefficient

We start by factoring out the leading coefficient (which is 2 in this case) from the first two terms:

y = 2(x² + 2x) + 7

Step 2: Complete the Square

Next, we need to complete the square for the expression inside the parentheses. To do this, we take the coefficient of x (which is 2), divide it by 2 (resulting in 1), and then square it (resulting in 1) as well. We add and subtract this square inside the parentheses:

y = 2(x² + 2x + 1 - 1) + 7

This simplifies to:

y = 2((x + 1)² - 1) + 7

Distributing the 2 gives us:

y = 2(x + 1)² - 2 + 7

Finally, we have:

y = 2(x + 1)² + 5

Step 3: Identify the Vertex

The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex. From our equation, we see that:

– The vertex is at (-1, 5), which means:

  • The minimum value of the function, since a = 2 (a positive value), occurs at y = 5.
  • The minimum point happens when x = -1.

Conclusion

Thus, by completing the square, we determine that the quadratic function y = 2x² + 4x + 7 reaches its minimum value of 5 at x = -1.

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