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Chapter 2: Problem 59

Use transformations to graph each function and state the domain and range. $$y=-2 \sqrt{x+3}+2$$

Short Answer

Expert verified
Domain: \[ [-3, \infty) \], Range: \[ (-\infty, 2] \]

Step by step solution

01

- Identify the base function

The base function is \( y = \sqrt{x} \).
02

- Apply the horizontal shift

The function \( y = \sqrt{x+3} \) represents a horizontal shift 3 units to the left.
03

- Apply the vertical stretch and reflection

The function \( y = -2 \sqrt{x+3} \) stretches the base function vertically by a factor of 2 and reflects it across the x-axis.
04

- Apply the vertical shift

The function \( y = -2 \sqrt{x+3} + 2 \) shifts the entire graph up 2 units.
05

- Determine the domain

The domain of the function is determined by the expression inside the square root. Since \( x + 3 \) must be non-negative, \( x \ge -3 \), so the domain is \[ [-3, \infty) \].
06

- Determine the range

Considering the transformations applied, the range can be determined by the highest and lowest points of the transformed function: \[ (-\infty, 2] \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
To understand the domain and range of the function, remember:
  • Domain represents all possible x-values of the function.
  • Range represents all possible y-values of the function.
For the given function, which includes a square root, the value inside the square root must be non-negative.Let鈥檚 start by examining the term inside the square root: \( x + 3 \).To keep this non-negative, \( x + 3 \ge 0 \), or simply \( x \ge -3 \).Thus, the domain is \[ [-3, \infty) \].For the range, consider the transformations applied:- The function is reflected downward and stretched by 2 units.- Finally, it is shifted up by 2 units.As a result, the highest point of the function is 2, and it extends downward infinitely.Thus, the range is \[(-\infty, 2] \].
Horizontal Shift
A horizontal shift occurs when we adjust the input value of the function. Given the function \( y = \sqrt{x+3} \), the term \( +3 \) suggests a horizontal shift.
  • This means the graph of the base function \( y = \sqrt{x} \) shifts to the left by 3 units.
Why to the left? Because adding to the x-value moves the graph in the negative x-direction.Think of it as getting the same y-value for an x-value that is smaller by 3. Therefore, each point on the graph of \( y = \sqrt{x} \) moves 3 units left to create \( y = \sqrt{x+3} \).
Vertical Stretch and Reflection
The transformations \( y = -2 \sqrt{x+3} \) include both vertical stretch and reflection. Here鈥檚 how:
  • Vertical Stretch: The coefficient 2 means the graph is stretched vertically by a factor of 2.
  • Reflection: The negative sign in front of 2 indicates a reflection across the x-axis.
Combined, this transforms the function \( \sqrt{x+3} \) by making every y-value twice as large and flipping it over the x-axis.For example, if \( \sqrt{x+3} \) outputs 1 for some x, \( -2 \sqrt{x+3} \) would output -2.
Vertical Shift
The transformation \( y = -2 \sqrt{x+3} + 2 \) includes a vertical shift. Here, the entire graph is shifted upwards by 2 units.
  • This means every y-value of \( -2 \sqrt{x+3} \) is increased by 2.
Imagine you draw the graph first without the vertical shift, then move each point up by 2 units. This vertical shift ensures that the final graph is elevated.Together with other transformations, the highest point of \( y = -2 \sqrt{x+3} + 2 \) is 2, and the graph extends downwards infinitely.

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