Write \( g(x)=-16 x+x^{2} \) in vertex form. \( \begin{array}{ll}\text { 1. Write the function in standard form. } & g(x)=x^{2}-16 x \\ \text { 2. Form a perfect square trinomial by adding and } & b=-16 \text {, so }\left(\frac{-16}{2}\right)^{2}=64 \\ \text { subtracting }\left(\frac{b}{2}\right)^{2} & g(x)=\left(x^{2}-16 x+64\right)-64 \\ \text { 3. Write the trinomial as a binomial squared. } & g(x)=(x-\square \\ \text { 4. Write the function is in vertex form, if needed. }\end{array} \)

To rewrite \( g(x) = -16 x + x^2 \) in vertex form, we start with the expression in standard form, which is \( g(x) = x^2 - 16x \). By completing the square, we add and then subtract 64 (since \( \left(-\frac{16}{2}\right)^2 = 64 \)). This gives us \( g(x) = (x^2 - 16x + 64) - 64 \), which simplifies to \( g(x) = (x - 8)^2 - 64 \). Thus, the vertex form of the function is \( g(x) = -(x - 8)^2 - 64 \). Knowing how to convert functions into vertex form can significantly aid in graphing quadratic functions. The vertex form highlights key features like the vertex, which is the minimum or maximum point of the parabola, depending on whether it opens upwards or downwards. For this function, the vertex is at the point (8, -64), making it super easy to plot and understand the graph's behavior! A common mistake when completing the square is forgetting to modify the original function appropriately when adding a constant. It’s essential to both add and subtract the square of half the coefficient of \( x \) to ensure you are not changing the equation’s value. For instance, neglecting the subtraction of 64 here would lead to miscalculating the vertex and skewing the function's overall shape! Remember, balance is key!