Question

Let g(x) be the transformation of f(x) = IxI such that the vertex is at (2, 5). Identify the rule for g(x) and its graph.

Answer 1

The image shows a graph of the absolute value function, f(x)=∣x∣, with a shift of two units to the right and five units up. This can be represented by the function g(x)=∣x−2∣+5.

The graph of the absolute value function, f(x)=∣x∣, is a V-shaped graph that is symmetrical about the y-axis. The vertex of the graph is at the origin, and the graph has two branches, one that extends to the left and one that extends to the right.

The graph of the function g(x)=∣x−2∣+5 is a transformation of the graph of f(x)=∣x∣. The graph of g(x) is shifted two units to the right and five units up. This means that the vertex of the graph of g(x) is at the point (2,5).

To see how the graph of g(x) is related to the graph of f(x), we can consider the following:

If x<2, then ∣x−2∣=−(x−2), so g(x)=−(x−2)+5=−x+7. This means that the graph of g(x) is reflected across the line x=2.

If x≥2, then ∣x−2∣=x−2, so g(x)=x−2+5=x+3. This means that the graph of g(x) is shifted two units to the right.

The following graph shows the graphs of f(x)=∣x∣ and g(x)=∣x−2∣+5:

Image of graphs of f(x) = x and g(x) = x2 + 5 is attached wsith the answer.

The graph of the function g(x)=∣x−2∣+5 is a transformation of the graph of the absolute value function, f(x)=∣x∣. The graph of g(x) is shifted two units to the right and five units up.

Answer 2

First remember that we have the initial function f(x)=|x| and we make 2 movements to the function in order to obtain g(x), lets's see what are those movements:

As you can see if we subtract -2 from the inside of the absolute value we move graph 2 positions to the right, now lets sum 5 to the function and see what happens

If we sum 5 to the function from outside the absolute value we move the graph 5 units up.

Se the answer to the question is the first option.