Question

The graphs for the functions f(x) = √x−3 and g(x) = √x−3 are shown in the graph below. Describe the transformations of a parent function to obtain the graphs of f(x) and g(x).

Answer 1

In order to obtain the graphs of the functions f(x) = √(x) - 3 and g(x) = √(x - 3), we start with the parent function f(x) = √(x).

For the function f(x) = √(x) - 3:

• Horizontal Shift: The parent function √(x) is shifted 3 units to the right to obtain √(x - 3). This means that the entire graph is shifted horizontally to the right by 3 units.

• Vertical Shift: After the horizontal shift, we subtract 3 from the function, resulting in √(x - 3) - 3. This vertical shift moves the entire graph downward by 3 units.

Therefore, the graph of f(x) = √(x) - 3 is obtained by shifting the parent function √(x) three units to the right and three units downward.

For the function g(x) = √(x - 3):

• Horizontal Shift: The parent function √(x) is shifted 3 units to the right to obtain √(x - 3). This horizontal shift moves the entire graph to the right by 3 units.
• No Vertical Shift: Unlike f(x), there is no vertical shift in g(x). The function remains at the same vertical level as the parent function.

Therefore, the graph of g(x) = √(x - 3) is obtained by shifting the parent function √(x) three units to the right, without any vertical shift.