Identify the graph of [tex]f(x)=3 \sqrt{x-2}+2[/tex].
Answer (2)
The function is a transformation of x with domain x ≥ 2 .
The graph starts at the point ( 2 , 2 ) .
The function increases as x increases.
The graph represents a square root function shifted right by 2, stretched vertically by 3, and shifted up by 2.
Explanation
Understanding the Function We are asked to identify the graph of the function f ( x ) = 3 x − 2 + 2 . This is a transformation of the square root function.
Finding the Domain First, let's determine the domain of the function. Since we have a square root, the expression inside the square root must be non-negative: x − 2 ≥ 0 . This means x ≥ 2 . So, the domain of the function is x ∈ [ 2 , ∞ ) .
Determining the Starting Point Next, let's find the starting point of the graph. The smallest value of x in the domain is x = 2 . When x = 2 , we have f ( 2 ) = 3 2 − 2 + 2 = 3 0 + 2 = 0 + 2 = 2 . So, the graph starts at the point ( 2 , 2 ) .
Analyzing the Function's Behavior Now, let's analyze the behavior of the function as x increases. Since the coefficient of the square root term is positive (3), the function will increase as x increases. This means the graph will go upwards as we move to the right.
Identifying the Transformations The function f ( x ) = 3 x − 2 + 2 is a transformation of the basic square root function y = x . The transformations are:
Horizontal shift to the right by 2 units (due to x − 2 ).
Vertical stretch by a factor of 3 (due to 3 x − 2 ).
Vertical shift upwards by 2 units (due to + 2 ).
Identifying the Graph Based on the domain, starting point, and increasing behavior, we can identify the correct graph. The graph should start at ( 2 , 2 ) and increase as x increases.
Examples
Understanding transformations of functions like f ( x ) = 3 x − 2 + 2 is useful in various fields. For example, in physics, the velocity of an object under constant acceleration can be modeled using a square root function. The transformations can represent changes in initial velocity or the magnitude of acceleration. In economics, a production function might involve square roots, and transformations could represent changes in technology or resource availability. Identifying the correct graph helps visualize and interpret these models.
The graph of the function f ( x ) = 3 x − 2 + 2 starts at the point ( 2 , 2 ) and increases as x increases. It represents a square root function that is transformed by shifting right by 2 units, stretching vertically by 3, and shifting up by 2. The domain of the function is [ 2 , ∞ ) .
;
