How do you rewrite the function g(x) = 4x^2 + 88x in vertex form?

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To rewrite the function g(x) = 4x² + 88x in vertex form, we will complete the square. The vertex form of a quadratic function is generally written as:

g(x) = a(x – h)² + k

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

Let’s go through the steps:

  1. Start with the given function: g(x) = 4x² + 88x.
  2. First, factor out the coefficient of x² from the first two terms:

    3. g(x) = 4(x² + 22x)

  3. Next, we need to complete the square inside the parentheses. To do this, take the coefficient of x (which is 22), divide it by 2 (getting 11), and square it (resulting in 121).
  4. Add and subtract this square inside the parentheses:

    5. g(x) = 4(x² + 22x + 121 – 121)

  5. This simplifies to:

    6. g(x) = 4((x + 11)² – 121)

  6. Now, distribute the 4:

    7. g(x) = 4(x + 11)² – 484

At this point, we have expressed the function g(x) in vertex form:

g(x) = 4(x + 11)² – 484

This shows that the vertex of the function is at the point (-11, -484).

This form provides insights into the function’s properties, such as its vertex, direction of opening, and the steepness of the parabola.

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