Identify the graph of [tex]f(x)=2 \sqrt{x-1}+4[/tex].
Answer (2)
The function is a transformation of the square root function.
The domain of the function is x ≥ 1 .
The graph starts at the point ( 1 , 4 ) .
The graph is shifted 1 unit to the right, stretched vertically by a factor of 2, and shifted 4 units up.
Explanation
Analyze the function We are given the function f ( x ) = 2 x − 1 + 4 and we need to identify its graph. Let's analyze the function to understand its properties and how it transforms the basic square root function.
Determine 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. So, x − 1 ≥ 0 , which means x ≥ 1 . Thus, the domain of the function is x ∈ [ 1 , ∞ ) .
Find the starting point Next, let's find the starting point of the graph. This occurs when x − 1 = 0 , which gives x = 1 . When x = 1 , we have f ( 1 ) = 2 1 − 1 + 4 = 2 0 + 4 = 0 + 4 = 4 . So, the starting point of the graph is ( 1 , 4 ) .
Analyze the transformations Now, let's analyze the transformations of the standard square root function y = x . The given function is f ( x ) = 2 x − 1 + 4 . This represents a horizontal shift of 1 unit to the right (due to x − 1 ), a vertical stretch by a factor of 2 (due to the 2 multiplying the square root), and a vertical shift of 4 units up (due to the +4).
Analyze the behavior of the function As x increases from 1, the value of x − 1 also increases. Since the square root is multiplied by 2 and then 4 is added, the function f ( x ) will also increase as x increases. The graph starts at ( 1 , 4 ) and increases as x increases.
Conclusion Based on the analysis, the graph of f ( x ) = 2 x − 1 + 4 is a square root function that starts at the point ( 1 , 4 ) and increases as x increases. It is shifted 1 unit to the right and 4 units up, and vertically stretched by a factor of 2.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how shifting and scaling affects wave functions is essential. Similarly, in economics, understanding how changes in parameters affect supply and demand curves is important. The transformations of the square root function can be applied in real-world scenarios such as modeling the growth of a population or the spread of a disease, where the rate of growth decreases over time.
The graph of the function f ( x ) = 2 x − 1 + 4 starts at the point ( 1 , 4 ) and increases as x increases. It undergoes a horizontal shift of 1 unit to the right, a vertical stretch by a factor of 2, and a vertical shift of 4 units up. The domain of the function is x ≥ 1 .
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