Which Represents A Stretch By A Factor Of 2 And A Vertical Shift Of 3 Units Down Of The Function $f(x)=\frac{1}{x}$?A. $f(x)=\frac{2}{x}-3$B. \$f(x)=\frac{3}{x+2}-3$[/tex\]C. $f(x)=\frac{2}{x-3}$D.
In mathematics, functions can be transformed in various ways to create new functions. These transformations can include stretching, shrinking, reflecting, and shifting the original function. In this article, we will focus on a specific type of transformation, which involves stretching a function by a factor of 2 and shifting it 3 units down.
Understanding the Original Function
The original function is given by $f(x)=\frac{1}{x}$. This is a classic example of a reciprocal function, which has a graph that resembles a hyperbola. The function has a vertical asymptote at , and as approaches infinity, the function approaches 0.
Stretching the Function
To stretch the function by a factor of 2, we need to multiply the function by 2. This means that the new function will have a vertical stretch of 2 units for every 1 unit of the original function. The new function can be represented as $f(x)=2\left(\frac{1}{x}\right)$.
Shifting the Function
To shift the function 3 units down, we need to subtract 3 from the function. This means that the new function will have a vertical shift of 3 units down from the original function. The new function can be represented as $f(x)=2\left(\frac{1}{x}\right)-3$.
Comparing the Options
Now that we have the transformed function, let's compare it with the options given:
A. $f(x)=\frac{2}{x}-3$ B. $f(x)=\frac{3}{x+2}-3$ C. $f(x)=\frac{2}{x-3}$ D. $f(x)=\frac{2}{x}+3$
Analyzing Option A
Option A is $f(x)=\frac{2}{x}-3$. This function has a vertical stretch of 2 units for every 1 unit of the original function, but it does not have a vertical shift of 3 units down. Instead, it has a vertical shift of 3 units up.
Analyzing Option B
Option B is $f(x)=\frac{3}{x+2}-3$. This function has a vertical shift of 3 units down, but it does not have a vertical stretch of 2 units for every 1 unit of the original function. Instead, it has a horizontal shift of 2 units to the left.
Analyzing Option C
Option C is $f(x)=\frac{2}{x-3}$. This function has a vertical stretch of 2 units for every 1 unit of the original function, but it does not have a vertical shift of 3 units down. Instead, it has a horizontal shift of 3 units to the right.
Analyzing Option D
Option D is $f(x)=\frac{2}{x}+3$. This function has a vertical stretch of 2 units for every 1 unit of the original function, but it does not have a vertical shift of 3 units down. Instead, it has a vertical shift of 3 units up.
Conclusion
Based on our analysis, the correct answer is option A, $f(x)=\frac{2}{x}-3$. This function has a vertical stretch of 2 units for every 1 unit of the original function and a vertical shift of 3 units down.
Final Answer
In our previous article, we discussed how to transform a function by stretching it by a factor of 2 and shifting it 3 units down. We also compared the options given to find the correct answer. In this article, we will provide a Q&A section to help you better understand the concept of transforming functions.
Q: What is the original function?
A: The original function is given by $f(x)=\frac{1}{x}$. This is a classic example of a reciprocal function, which has a graph that resembles a hyperbola.
Q: What is the effect of stretching the function by a factor of 2?
A: When we stretch the function by a factor of 2, we multiply the function by 2. This means that the new function will have a vertical stretch of 2 units for every 1 unit of the original function.
Q: What is the effect of shifting the function 3 units down?
A: When we shift the function 3 units down, we subtract 3 from the function. This means that the new function will have a vertical shift of 3 units down from the original function.
Q: How do we combine the stretch and shift transformations?
A: To combine the stretch and shift transformations, we multiply the function by 2 and then subtract 3. This gives us the new function $f(x)=2\left(\frac{1}{x}\right)-3$.
Q: What is the correct answer among the options given?
A: The correct answer is option A, $f(x)=\frac{2}{x}-3$. This function has a vertical stretch of 2 units for every 1 unit of the original function and a vertical shift of 3 units down.
Q: What are some common types of function transformations?
A: Some common types of function transformations include:
- Stretching: multiplying the function by a constant factor
- Shrinking: dividing the function by a constant factor
- Reflecting: flipping the function over a horizontal or vertical axis
- Shifting: moving the function up or down, left or right
Q: How do we determine the type of transformation that has occurred?
A: To determine the type of transformation that has occurred, we need to analyze the function and look for changes in its graph. We can use the following steps:
- Identify the original function
- Compare the original function with the new function
- Look for changes in the graph, such as vertical or horizontal shifts, stretches or shrinks, or reflections
Q: What are some real-world applications of function transformations?
A: Function transformations have many real-world applications, including:
- Physics: to describe the motion of objects
- Engineering: to design and optimize systems
- Economics: to model economic systems and make predictions
- Computer Science: to develop algorithms and data structures
Conclusion
In this article, we provided a Q&A section to help you better understand the concept of transforming functions. We discussed the original function, the effect of stretching and shifting, and how to combine these transformations. We also provided some common types of function transformations and real-world applications. We hope this article has been helpful in your understanding of function transformations.