Consider the functions:
[tex]
\begin{array}{l}
f(x)=\sqrt{x} \\
g(x)=\sqrt{x-3}+1 \\
h(x)=\sqrt{x+1}-2
\end{array}
[/tex]
Answer (1)
g ( x ) is obtained from f ( x ) by shifting the graph 3 units to the right and 1 unit upward.
h ( x ) is obtained from f ( x ) by shifting the graph 1 unit to the left and 2 units downward.
The transformations involve horizontal and vertical shifts of the parent function f ( x ) = x .
The transformations are f ( x ) → g ( x ) and f ( x ) → h ( x ) .
Explanation
Understanding the Functions We are given three functions: f ( x ) = x , g ( x ) = x − 3 + 1 , and h ( x ) = x + 1 − 2 .
Our goal is to describe how g ( x ) and h ( x ) are transformations of the parent function f ( x ) .
Analyzing g(x) The function g ( x ) is derived from f ( x ) through two transformations: a horizontal shift and a vertical shift. Specifically, g ( x ) = x − 3 + 1 indicates that we replace x in f ( x ) with x − 3 , which shifts the graph 3 units to the right. Then, we add 1 to the entire expression, which shifts the graph 1 unit upward.
Analyzing h(x) Similarly, h ( x ) is also derived from f ( x ) through horizontal and vertical shifts. The function h ( x ) = x + 1 − 2 shows that we replace x in f ( x ) with x + 1 , which shifts the graph 1 unit to the left. Then, we subtract 2 from the entire expression, which shifts the graph 2 units downward.
Conclusion In summary, g ( x ) is obtained from f ( x ) by shifting the graph 3 units to the right and 1 unit upward. The function h ( x ) is obtained from f ( x ) by shifting the graph 1 unit to the left and 2 units downward.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions helps predict the trajectory of objects. Similarly, in economics, transformations of supply and demand curves can model the impact of taxes or subsidies. By recognizing these transformations, we can better understand and predict real-world phenomena.
