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Magazine Chapter 8: Problem 18
Sketch the graph of each function. $$ f(x)=\sqrt{x+3}+2 $$
Short Answer
Expert verified
Graph shifts the square root function left 3 units and up 2 units, starting at (-3,2).
Step by step solution
01
Identify the Basic Shape
The function \( f(x) = \sqrt{x+3} + 2 \) is based on the square root function \( g(x) = \sqrt{x} \). The basic shape of \( g(x) \) is a curve that starts at the origin (0,0) and increases slowly as \( x \) increases.
02
Determine the Domain
For the function \( f(x) = \sqrt{x+3} + 2 \), the expression inside the square root \( x+3 \) must be greater than or equal to 0. Thus, the domain of \( f \) is \( x \geq -3 \). This means that the graph starts at \( x = -3 \) and extends to the right indefinitely.
03
Apply Horizontal Shift
The term \( x+3 \) inside the square root indicates a horizontal shift to the left by 3 units compared to \( g(x) = \sqrt{x} \). Start the graph at \( (-3,0) \) instead of the origin.
04
Apply Vertical Shift
The \(+2\) outside the square root shifts the entire graph of \( y = \sqrt{x+3} \) up by 2 units. Therefore, move each point on the basic square root graph up by 2 units. This means the starting point is now \((-3, 2)\).
05
Plot Key Points and Draw the Graph
Determine some key points by substituting convenient values of \( x \) into \( f(x) \), such as \( x = -3, 0, 1 \). For example, at \( x = -3 \), \( f(-3) = \sqrt{-3+3} + 2 = 2 \). At \( x = 0 \), \( f(0) = \sqrt{3} + 2 \), which is approximately 3.73. Plot these key points and draw a smooth curve passing through them.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Functions
Graphing functions is like drawing a picture of the relationships between variables. When graphing, the aim is to visualize how the output of a function changes as the input changes. For every function, there is a graph that represents it, giving us a deeper understanding of its behavior.
Here's a simple approach to graphing:
Here's a simple approach to graphing:
- Identify the basic function shape. Every function type, like linear, quadratic, or square root, has a distinct shape.
- Determine the domain of the function. This tells you the range of input values you can use.
- Apply transformations like shifting and stretching to modify the basic shape into the graph of your function.
- Plot key points. These include intercepts and easily calculated values to guide your graph drawing.
- Draw the curve smoothly to capture the continuous nature of the function.
Square Root Function
The square root function is a special type of function characterized by the expression \( g(x) = \sqrt{x} \). This function has a unique, gentle curvature that begins at the origin and rises slowly as \( x \) increases.
Some important traits of the square root function include:
Some important traits of the square root function include:
- Its domain is limited. \( x \) must be greater than or equal to zero because we cannot take the square root of a negative number in the set of real numbers.
- The basic shape starts from the point (0,0) and rises to the right.
- It is non-linear, meaning its graph is a curve, not a straight line.
- As \( x \) gets larger, the rate of increase in \( g(x) \) becomes slower.
Function Transformations
Function transformations are techniques used to alter the graph of a function to create a new function. These transformations include translations (shifts), reflections, rotations, and dilations (stretches and compressions).
In the case of the function \( f(x) = \sqrt{x+3} + 2 \), we apply transformations to the square root function \( g(x) = \sqrt{x} \):
In the case of the function \( f(x) = \sqrt{x+3} + 2 \), we apply transformations to the square root function \( g(x) = \sqrt{x} \):
- Horizontal Shift: The term \( x+3 \) signifies a shift to the left by 3 units. If it were \( x-3 \), the shift would be to the right.
- Vertical Shift: The addition of 2 outside the square root, \( +2 \), raises the entire graph upwards by 2 units.
- These shifts adjust the starting point of the graph, moving it from (0,0) to (-3,2).
