Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Use transformation, showing all work. Use interval notation. Problem attached

**Graphing Transformed Functions** In this activity, we will use transformations to graph the function and state the domain and range using interval notation. The function we will be working with is: \[ y = -\sqrt{x + 2} + 1 \] **Steps and Transformations:** 1. **Base Function:** - The basic function here is \( y = -\sqrt{x} \), which is a reflection of the square root function \( y = \sqrt{x} \) across the x-axis. 2. **Horizontal Shift:** - The expression \( x + 2 \) indicates a horizontal shift to the left by 2 units. This transformation moves the graph of \( y = -\sqrt{x + 2} \). 3. **Vertical Shift:** - The addition of \( +1 \) at the end of the function raises the graph vertically by 1 unit. So, we get \( y = -\sqrt{x + 2} + 1 \). Using the transformations, we sketch the graph as follows: **Graphs and Diagrams Explanation:** - The coordinate plane is labeled with both x and y axes ranging from -5 to 5. - The graph starts at \( (-2, 1) \) since at \( x = -2 \), \( \sqrt{0} = 0 \) and \( y = 1 \). - As \( x \) increases, the function \( y = -\sqrt{x + 2} + 1 \) decreases because of the negative sign before the square root, reflecting the curve downward. **Domain and Range:** - The **domain** is the set of all x-values that can be input into the function. Since the inside of the square root \( x + 2 \) must be non-negative, the domain is: \[ x \ge -2 \quad \text{or in interval notation:} \quad [-2, \infty) \] - The **range** is the set of all possible y-values of the function. Given the transformations applied: - The highest point on the graph is at \( y = 1 \). - As \( x \) increases, \( y \) will continue to decrease indefinitely. Hence, the range is: \[ y \le 1 \quad \text{or in interval notation:} \quad (-
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Transcribed Image Text:**Graphing Transformed Functions** In this activity, we will use transformations to graph the function and state the domain and range using interval notation. The function we will be working with is: \[ y = -\sqrt{x + 2} + 1 \] **Steps and Transformations:** 1. **Base Function:** - The basic function here is \( y = -\sqrt{x} \), which is a reflection of the square root function \( y = \sqrt{x} \) across the x-axis. 2. **Horizontal Shift:** - The expression \( x + 2 \) indicates a horizontal shift to the left by 2 units. This transformation moves the graph of \( y = -\sqrt{x + 2} \). 3. **Vertical Shift:** - The addition of \( +1 \) at the end of the function raises the graph vertically by 1 unit. So, we get \( y = -\sqrt{x + 2} + 1 \). Using the transformations, we sketch the graph as follows: **Graphs and Diagrams Explanation:** - The coordinate plane is labeled with both x and y axes ranging from -5 to 5. - The graph starts at \( (-2, 1) \) since at \( x = -2 \), \( \sqrt{0} = 0 \) and \( y = 1 \). - As \( x \) increases, the function \( y = -\sqrt{x + 2} + 1 \) decreases because of the negative sign before the square root, reflecting the curve downward. **Domain and Range:** - The **domain** is the set of all x-values that can be input into the function. Since the inside of the square root \( x + 2 \) must be non-negative, the domain is: \[ x \ge -2 \quad \text{or in interval notation:} \quad [-2, \infty) \] - The **range** is the set of all possible y-values of the function. Given the transformations applied: - The highest point on the graph is at \( y = 1 \). - As \( x \) increases, \( y \) will continue to decrease indefinitely. Hence, the range is: \[ y \le 1 \quad \text{or in interval notation:} \quad (-
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