Problem 55 - 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. $$g(x)=(x-2)^{2}$$

Step-by-Step Solution

Answer

The graph of function \(g(x)=(x-2)^{2}\) is a standard quadratic function shifted \(2\) units to the right from the vertex \((0, 0)\) to \((2, 0)\).

Step 1: Graph the standard quadratic function

Begin by graphing the standard quadratic function \(f(x)=x^{2}\). This is a parabolic graph where the vertex is at the origin \((0, 0)\) and it opens upwards.

Step 2: Identify the transformation

In the transformed function \(g(x)=(x-2)^{2}\), we can see the quadratic function is not different, but an alteration on the x variable has been made. The form \((x-2)\) indicates the graph should be shifted 2 units to the right, as it comes in the form \((x-h)\) where \(h\) corresponds to the horizontal shift.

Step 3: Graph the transformed function

Starting from the original function graph \(f(x)=x^{2}\), apply a shift of 2 units to the right to every point to obtain the transfromed function graph \(g(x) = (x-2)^2\). This moves the vertex of the parabolic graph to the point \((2, 0)\) and it still opens upwards.