Problem 55 - College Algebra

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 the function \(g(x)=(x-2)^2\) is a parabola opening upwards, identical in shape to graph of \(f(x) = x^2\), but shifted 2 units to the right.

Step 1: Graph the Standard Quadratic Function

The standard quadratic function is \(f(x) = x^2\). Draw a graph of this function. It will form a parabola opening upwards with vertex at origin i.e. point (0, 0).

Step 2: Identify and Understand the Transformation

The function \(g(x)=(x-2)^2\) is a transformation of the standard quadratic function \(f(x) = x^2\). Here, \(x\) in \(f(x)\) is replaced by \((x-2)\). This represents a horizontal shift, or a translation of the graph along x-axis. The entire graph of \(f(x)\) will shift 2 units to right because of the transformation \(x-2\) gives in function \(g(x)\)

Step 3: Apply the Transformation and Graph the Transformed Function

Now, draw the graph of the function \(g(x)\) by shifting the graph of \(f(x)\) 2 units to the right. This moves the vertex of the parabola from origin to the point (2, 0)