12. \( g(x)=(x-1)^{2} \)

Did you know that quadratic functions like \( g(x) = (x-1)^{2} \) graph into beautiful parabola shapes? This specific parabola opens upwards and has its vertex at the point (1, 0). The vertex is the lowest point on the graph, so any value of \( x \) will make \( g(x) \) greater than or equal to zero, creating a lovely symmetric shape! When working with functions like this, a common mistake is forgetting to check the vertex and intercepts. To find the x-intercept, set \( g(x) = 0 \). Here, that gives us just one solution: \( x = 1 \). Always remember, especially with quadratics, that there can be unique characteristics like a double root, which may not be immediately obvious!