Problem 56 - College Algebra Essentials

Question

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$g(x)=(x-1)^{2}$$

Step-by-Step Solution

Answer

The function \(g(x)=(x-1)^{2}\) is a horizontal shift of the standard quadratic function \(f(x)=x^{2}\) to the right by one unit.

Step 1: Graph the Standard Quadratic function

Start by graphing the standard quadratic function \(f(x)=x^{2}\). The standard quadratic function is a parabola that opens upwards. The vertex of the graph is at the origin (0,0). It is advised to plot more points for a thorough graph, for instance take x = -2, -1, 0, 1, 2. Calculate the corresponding y values, then plot these points and draw the graph.

Step 2: Understand the transformation

The given transformed function is \(g(x)=(x-1)^{2}\). This is a horizontal shift of 'f(x)' to the right by 1 step. In general, for a function \(f(x)\), a function of the form \(f(x-h)\) represents a shift of 'h' units to the right.

Step 3: Graph the transformed function

Next, graph the function \(g(x)=(x-1)^{2}\). The new vertex of the graph will be (1,0). Keep the shape of the graph the same as the standard quadratic function and shift it one unit to the right. Again, it could be useful to plot multiple points. For this function, choosing x values such as -1, 0, 1, 2, 3 could be done, with the associated y values calculated and thereafter plotted on the graph. This plot should clearly depict the shift from the original function.