How do you convert the quadratic function y = x^2 + 4x + 7 into vertex form?

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To convert the quadratic function from its standard form, y = x2 + 4x + 7, into vertex form, you need to complete the square. The vertex form of a quadratic function is given by:

y = a(x – h)2 + k,

where (h, k) is the vertex of the parabola.

Here are the steps to follow:

  1. Start with the original function: y = x2 + 4x + 7
  2. Isolate the quadratic and linear terms: y = (x2 + 4x) + 7
  3. Complete the square:
    • Take the coefficient of x (which is 4), divide it by 2, and square it: (4/2)2 = 4.
    • Add and subtract this square inside the parentheses:

    y = (x2 + 4x + 4 – 4) + 7

  4. Rewrite the equation: Factor the perfect square trinomial and simplify:

    5. y = (x + 2)2 – 4 + 7

    y = (x + 2)2 + 3

  5. Final vertex form: Thus, the function in vertex form is:

    6. y = (x + 2)2 + 3

The vertex of the parabola is at the point (-2, 3).

In conclusion, the quadratic function in vertex form is y = (x + 2)2 + 3.

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