Identify the graph of [tex]f(x)=\sqrt{x-2}+3[/tex].
Answer (2)
The function f ( x ) = s q r t x − 2 + 3 is a square root function with a domain of x g e q 2 . The graph starts at the point ( 2 , 3 ) and increases as x increases. The graph is a transformation of y = s q r t x , shifted right by 2 and up by 3.
Explanation
Understanding the Function We are given the function f ( x ) = s q r t x − 2 + 3 and asked to identify its graph.
Determining the Domain The square root function x is only defined for non-negative values, so we must have x − 2 g e q 0 , which means x g e q 2 . This tells us the domain of the function is x g e q 2 .
Finding the Starting Point When x = 2 , we have f ( 2 ) = s q r t 2 − 2 + 3 = s q r t 0 + 3 = 0 + 3 = 3 . So the graph starts at the point ( 2 , 3 ) .
Analyzing the Graph's Behavior As x increases from 2, the value of x − 2 increases, so x − 2 increases, and thus f ( x ) = s q r t x − 2 + 3 also increases. The graph will start at ( 2 , 3 ) and increase as x increases.
Identifying Transformations The graph of f ( x ) is a transformation of the basic square root function y = s q r t x . The transformation involves a horizontal shift to the right by 2 units and a vertical shift upwards by 3 units.
Conclusion Based on the analysis, the graph should start at the point ( 2 , 3 ) and increase as x increases, existing only for x g e q 2 . Therefore, we need to choose the graph that satisfies these conditions.
Examples
Understanding transformations of functions like f ( x ) = s q r t x − 2 + 3 is useful in many real-world applications. For example, in physics, the velocity of an object might be modeled by a square root function, and understanding how shifts affect the graph can help predict the object's behavior under different conditions. Similarly, in economics, a cost function might involve a square root, and transformations can help analyze how changes in production affect costs.
The graph of the function f ( x ) = x − 2 + 3 starts at the point ( 2 , 3 ) and increases for x ≥ 2 . It is a transformation of the standard square root function, shifted right by 2 units and up by 3 units. Thus, the graph will be in the first quadrant and only exist for values of x greater than or equal to 2.
;
