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Magazine Chapter 8: Problem 29
Sketch the graph of function. $$f(x)=\sqrt{x-2}+3$$
Short Answer
Expert verified
The graph of \(f(x) = \sqrt{x-2} + 3\) starts at (2,3) and moves right, following the typical square root shape.
Step by step solution
01
Identify the Function Type
The function given is \[f(x) = \sqrt{x-2} + 3\]This is a square root function, which is a transformation of the basic square root function \(\sqrt{x}\). The transformations include a horizontal shift and a vertical shift.
02
Determine the Domain
The domain of \(f(x) = \sqrt{x-2} + 3\) is determined by the expression under the square root, \(x - 2\), which must be greater than or equal to zero. Thus, we solve \(x - 2 \geq 0\) to find that the domain is \[x \geq 2\].
03
Understand the Transformations
This function can be seen as \(y = \sqrt{x}\) first undergoing a horizontal shift to the right by 2 units (since \(x - 2\)) and then a vertical shift upward by 3 units (since \(+ 3\)).
04
Sketch the Basic Function
Sketch the basic function \(y = \sqrt{x}\), which starts at (0,0) and passes through points like (1,1), (4,2) due to the square of those numbers.
05
Apply the Horizontal Shift
Shift the basic function \(y = \sqrt{x}\) to the right by 2 units. Each point \((a, b)\) on the graph of \(y = \sqrt{x}\) moves to \((a + 2, b)\). Example: (0,0) becomes (2,0), and (1,1) becomes (3,1).
06
Apply the Vertical Shift
Now shift the resulting function up by 3 units. Each point \((a, b)\) moves to \((a, b + 3)\). Example: (2,0) becomes (2,3), and (3,1) becomes (3,4).
07
Sketch the Final Graph
The graph of \(f(x) = \sqrt{x-2} + 3\) starts at (2,3) and moves rightwards, approaching points such as (3,4), (6,5), etc., by following the path of the square root curve shifted as described.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Transforming Functions
Function transformations are changes made to the basic form of a function, allowing us to visualize different aspects or forms of the same function. For the square root function given, \[f(x) = \sqrt{x - 2} + 3\]we see two primary types of transformations: horizontal and vertical shifts.
- Horizontal Shift: This occurs when we modify the expression inside the square root. In this case, the function \(\sqrt{x}\) becomes \(\sqrt{x - 2}\), indicating a shift 2 units to the right. To understand this, think of the function beginning at the x-value that makes the inside zero, hence shifting the graph to the right by that value.
- Vertical Shift: This occurs when we add or subtract a number outside the square root. Here, adding 3, as in \(\sqrt{x - 2} + 3\), moves the entire graph upwards by 3 units. Thus, every value of the original function is just 3 units higher.
Understanding the Domain of a Function
Determining the domain of a function is a crucial step before graphing it. The domain tells us what x-values are permissible, allowing us to understand where the graph of a function should exist along the x-axis.For the function \[f(x) = \sqrt{x - 2} + 3\], we need to ensure the expression under the square root, \(x - 2\), is non-negative. This is because you can't take the square root of a negative number in the realm of real numbers.
- This means solving the inequality \(x - 2 \geq 0\), giving us \(x \geq 2\).
- Therefore, the domain of this function is all x-values starting from 2 onwards: \[x \in [2, \infty)\]
Graph Sketching Techniques
Sketching graphs can seem daunting, but with systematic steps, it becomes easier. For our function \[f(x) = \sqrt{x - 2} + 3\], we start by sketching the basic square root function \(y = \sqrt{x}\). This basic graph starts at (0,0) and curves gently upwards.To transform this graph based on our function:
- First, perform a horizontal shift: Move the graph of \(y = \sqrt{x}\) 2 units to the right. This means a point like (0,0) becomes (2,0).
- Second, apply the vertical shift: Raise the entire graph 3 units up. Hence, (2,0) transforms to (2,3).
- Continue by moving each consequential point in this manner. The next important point, originally (1,1), becomes (3,4) after both shifts.
