Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Verified step by step guidance

1. Identify the base function: The graph resembles the shape of the square root function, so the base function is g(x) = \sqrt{x}.
2. Determine the horizontal shift: The graph does not appear to be shifted horizontally, so there is no change in the x-direction.
3. Determine the vertical shift: The graph appears to be shifted downward by 5 units. This means we subtract 5 from the base function.
4. Write the transformed function: Combine the base function and the vertical shift to get the transformed function g(x) = \sqrt{x} - 5.
5. Verify the transformation: Check if the graph matches the transformed function g(x) = \sqrt{x} - 5 by comparing key points and the general shape.

Key Concepts

Transformations of Functions

Transformations of functions involve altering the graph of a parent function through shifts, stretches, compressions, and reflections. Common transformations include vertical and horizontal shifts, which move the graph up, down, left, or right, and vertical stretches or compressions that change the steepness of the graph. Understanding these transformations is crucial for predicting how the graph of a function will change when its equation is modified.

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, produces a V-shaped graph that reflects all negative values of x to positive values. This function is essential in understanding how transformations affect its shape and position. When transformed, the graph can shift vertically or horizontally, or change its orientation, which can be analyzed through its equation.

Square Root Function

The square root function, represented as g(x) = √x, produces a graph that starts at the origin and increases gradually. This function is important for understanding transformations that can affect its domain and range. Transformations can include vertical shifts, which move the graph up or down, and horizontal shifts, which can change where the graph begins, impacting its overall appearance.