Let $f(x) = -2 \sqrt{x-3} - 2$. Make A Table Of Values By Finding The Values Of $y$ Given The Values Of $ X X X [/tex]. Use A Calculator. Plot The Points And Connect With A Smooth Curve. Describe The Transformation
1. Exploring the Function f(x) = -2√(x-3) - 2
Understanding the Function
The given function is f(x) = -2√(x-3) - 2. This function involves a square root and a linear term. To understand the behavior of this function, we need to create a table of values by finding the values of y given the values of x.
Creating a Table of Values
To create a table of values, we will use a calculator to find the values of y for different values of x. We will start by finding the values of y for x = 0, 1, 2, 3, 4, 5, and 6.
| x | y = f(x) = -2√(x-3) - 2 |
|---|---|
| 0 | -2√(-3) - 2 |
| 1 | -2√(-2) - 2 |
| 2 | -2√(-1) - 2 |
| 3 | -2√0 - 2 = -2 |
| 4 | -2√1 - 2 = -4 |
| 5 | -2√2 - 2 |
| 6 | -2√3 - 2 |
Plotting the Points and Connecting with a Smooth Curve
To plot the points, we will use a graphing calculator or a computer program. We will plot the points (x, y) for each value of x in the table.
After plotting the points, we will connect them with a smooth curve. The resulting graph will give us an idea of the behavior of the function.
Describing the Transformation
The graph of the function f(x) = -2√(x-3) - 2 is a transformation of the square root function. The square root function is given by f(x) = √x. To obtain the given function, we have made the following transformations:
- Horizontal Shift: The function has been shifted 3 units to the right. This is because the term (x-3) inside the square root has been added to x.
- Vertical Stretch: The function has been stretched vertically by a factor of 2. This is because the term -2 has been multiplied with the square root term.
- Reflection: The function has been reflected across the x-axis. This is because the term -2 has been added to the square root term.
Key Features of the Graph
The graph of the function f(x) = -2√(x-3) - 2 has the following key features:
- Domain: The domain of the function is all real numbers greater than or equal to 3.
- Range: The range of the function is all real numbers less than or equal to -2.
- Asymptote: The function has a horizontal asymptote at y = -2.
- Intercepts: The function has a y-intercept at (0, -2) and an x-intercept at (3, -2).
Conclusion
In this article, we have explored the function f(x) = -2√(x-3) - 2. We have created a table of values, plotted the points, and connected them with a smooth curve. We have also described the transformation of the function and identified its key features. The graph of the function is a transformation of the square root function, involving a horizontal shift, vertical stretch, and reflection.
2. Q&A: Exploring the Function f(x) = -2√(x-3) - 2
Frequently Asked Questions
In this article, we will answer some frequently asked questions about the function f(x) = -2√(x-3) - 2.
Q: What is the domain of the function f(x) = -2√(x-3) - 2?
A: The domain of the function is all real numbers greater than or equal to 3. This is because the term (x-3) inside the square root must be non-negative.
Q: What is the range of the function f(x) = -2√(x-3) - 2?
A: The range of the function is all real numbers less than or equal to -2. This is because the term -2 has been added to the square root term, which means that the function will always be less than or equal to -2.
Q: What is the horizontal asymptote of the function f(x) = -2√(x-3) - 2?
A: The horizontal asymptote of the function is y = -2. This is because as x approaches infinity, the term -2√(x-3) approaches -2.
Q: What is the y-intercept of the function f(x) = -2√(x-3) - 2?
A: The y-intercept of the function is (0, -2). This is because when x = 0, the term -2√(x-3) becomes -2.
Q: What is the x-intercept of the function f(x) = -2√(x-3) - 2?
A: The x-intercept of the function is (3, -2). This is because when y = 0, the term -2√(x-3) becomes -2, which means that x = 3.
Q: How do I graph the function f(x) = -2√(x-3) - 2?
A: To graph the function, you can use a graphing calculator or a computer program. You can also plot the points (x, y) for each value of x in the table and connect them with a smooth curve.
Q: What is the transformation of the function f(x) = -2√(x-3) - 2?
A: The transformation of the function is a horizontal shift of 3 units to the right, a vertical stretch by a factor of 2, and a reflection across the x-axis.
Q: What is the significance of the term -2 in the function f(x) = -2√(x-3) - 2?
A: The term -2 in the function represents a vertical stretch by a factor of 2 and a reflection across the x-axis.
Q: What is the significance of the term (x-3) in the function f(x) = -2√(x-3) - 2?
A: The term (x-3) in the function represents a horizontal shift of 3 units to the right.
Conclusion
In this article, we have answered some frequently asked questions about the function f(x) = -2√(x-3) - 2. We have discussed the domain, range, horizontal asymptote, y-intercept, x-intercept, and transformation of the function. We have also explained the significance of the terms -2 and (x-3) in the function.