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

Step-by-Step Solution

Answer

The graph of \(g(x)=(x-2)^{2}\) is a parabola opening upwards, with its vertex at the point (2, 0).

Step 1: Graph the standard quadratic function

The standard quadratic function is \(f(x) = x^{2}\). This makes a parabola, which is a u-shaped curve, that opens upwards. The vertex (lowest point) is at the origin (0, 0) and it is symmetric about the y-axis.

Step 2: Understand the transformation

The given function, \(g(x)=(x-2)^{2}\), is a transformation of the standard quadratic function. The (x-2) inside the function causes a horizontal shift to the right by 2 units. This means every point on the standard function f(x) = x2 will move 2 units right to form the function g(x)=(x-2)^2.

Step 3: Graph the transformed function

Start from the graph of the standard quadratic function. Then move each point 2 units to the right. This will result in a parabola that opens up, but its vertex is now at the point (2, 0), and it is still symmetric about the line x=2.