The function `g` is defined by `g(x) = x^2 - 8x + 7`. Which of the following equivalent forms of `g(x)` displays the co…

The question asks for an equivalent form of the function `g(x) = x² - 8x + 7` that shows the vertex coordinates as constants. This is known as the vertex form of a quadratic, which is `y = a(x - h)² + k`, where the point `(h, k)` is the vertex.

To convert the given function to vertex form, we can use the method of completing the square.

1. Start with the function `g(x) = x² - 8x + 7`.

2. To create a perfect square trinomial from `x² - 8x`, we take half of the coefficient of the `x` term (-8), which is -4, and square it, which gives 16.

3. We add and subtract 16 to the expression to keep its value the same: `g(x) = (x² - 8x + 16) - 16 + 7`.

4. The terms in the parenthesis form a perfect square: `(x² - 8x + 16) = (x - 4)²`.

5. Combine the constant terms: `-16 + 7 = -9`.

6. The resulting function is `g(x) = (x - 4)² - 9`.

This form displays the vertex coordinates, (4, -9), as constants in the expression.

Therefore, the correct choice is C.

A is incorrect. This is the factored form of the function. This form is equivalent to the given function, but it displays the x-intercepts of the graph (1 and 7) as constants, not the vertex coordinates.

B is incorrect. This form likely results from an error in completing the square. After adding 16 to create the perfect square, one must subtract 16 to keep the expression equivalent. This answer results from incorrectly adding 16 to the constant term 7 (`7 + 16 = 23`).

D is incorrect. This form likely results from a sign error. The x-coordinate of the vertex is 4, which means the term in the vertex form should be `(x - 4)²`, not `(x + 4)²`.