Problem 56 - Precalculus 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 graph of the given quadratic function \( g(x)=(x-1)^{2} \) is derived from the standard quadratic function \( f(x)=x^{2} \) by shifting the entire graph of \( f(x)=x^{2} \) 1 unit to the right. Thus, the given function will have a parabola-shape curve, symmetric with the y-axis, and with its vertex located at (1, 0).

Step 1: Identify the Standard Quadratic Function

The standard quadratic function is \( f(x)=x^{2} \). It is a parabola-shaped curve which is symmetric about the y-axis and has its lowest point at the origin.

Step 2: Identify the Given Function and Its Transformation

The given function is \( g(x)=(x-1)^{2} \). This is a transformation of the standard quadratic function \( f(x)=x^{2} \). Specifically, it is a horizontal shift of the standard function towards the right side by 1 unit.

Step 3: Graphing the Standard and Given Functions

Begin by plotting the standard function \( f(x)=x^{2} \) on the graph. This will be a curve symmetric about the y-axis with its vertex at the origin. Then, plot the graph of the given function \( g(x)=(x-1)^{2} \) by shifting the graph of \( f(x)=x^{2} \) 1 unit towards the right. This new graph will also be symmetric to the y-axis, but with its vertex at (1, 0).