The Function $f(x)=3 \cdot 4^x+5$ Is Translated To Become $g(x)=3 \cdot 4^x-2$. What Is The Effect On $ F ( X ) F(x) F ( X ) [/tex]?A. F ( X F(x F ( X ] Moves 7 Units Downward.B. F ( X F(x F ( X ] Moves 2 Units Downward.C.
Introduction
In mathematics, functions are used to describe the relationship between variables. When a function is translated, it means that the original function is shifted or moved to a new position on the coordinate plane. In this article, we will discuss the effect of translation on a function, specifically the function $f(x)=3 \cdot 4^x+5$, which is translated to become $g(x)=3 \cdot 4^x-2$.
What is Translation?
Translation is a type of transformation that involves moving a function to a new position on the coordinate plane. It can be horizontal, vertical, or a combination of both. In the case of the function $f(x)=3 \cdot 4^x+5$, it is translated vertically to become $g(x)=3 \cdot 4^x-2$.
Vertical Translation
Vertical translation involves moving a function up or down on the coordinate plane. When a function is translated vertically, the value of the function changes, but the shape of the function remains the same. In the case of the function $f(x)=3 \cdot 4^x+5$, it is translated vertically to become $g(x)=3 \cdot 4^x-2$.
Effect of Vertical Translation
When a function is translated vertically, the effect on the original function is a change in the value of the function. In the case of the function $f(x)=3 \cdot 4^x+5$, which is translated to become $g(x)=3 \cdot 4^x-2$, the effect is a downward shift of 7 units.
Why is the Effect a Downward Shift of 7 Units?
The effect of the vertical translation is a downward shift of 7 units because the original function $f(x)=3 \cdot 4^x+5$ has a value of 5, and the translated function $g(x)=3 \cdot 4^x-2$ has a value of -2. The difference between the two values is 7, which is the amount of the downward shift.
Conclusion
In conclusion, the effect of the translation on the function $f(x)=3 \cdot 4^x+5$ is a downward shift of 7 units. This is because the original function has a value of 5, and the translated function has a value of -2, resulting in a difference of 7 units.
Example
To illustrate the effect of the translation, let's consider an example. Suppose we have the function $f(x)=3 \cdot 4^x+5$, and we want to translate it to become $g(x)=3 \cdot 4^x-2$. The effect of the translation would be a downward shift of 7 units, resulting in the new function $g(x)=3 \cdot 4^x-2$.
Graphical Representation
The graphical representation of the function $f(x)=3 \cdot 4^x+5$ and the translated function $g(x)=3 \cdot 4^x-2$ is shown below:
| Function | Graph |
|---|---|
| $f(x)=3 \cdot 4^x+5$ |  |
| $g(x)=3 \cdot 4^x-2$ | |
As shown in the graphical representation, the function $f(x)=3 \cdot 4^x+5$ is translated vertically to become $g(x)=3 \cdot 4^x-2$, resulting in a downward shift of 7 units.
Key Takeaways
- The effect of translation on a function is a change in the value of the function.
- Vertical translation involves moving a function up or down on the coordinate plane.
- The effect of vertical translation is a downward shift of 7 units when the original function has a value of 5 and the translated function has a value of -2.
References
- [1] "Functions and Graphs" by Michael Sullivan
- [2] "Mathematics for the Nonmathematical" by Morris Kline
Discussion
Introduction
In our previous article, we discussed the effect of translation on a function, specifically the function $f(x)=3 \cdot 4^x+5$, which is translated to become $g(x)=3 \cdot 4^x-2$. In this article, we will answer some frequently asked questions about the effect of translation on a function.
Q: What is the effect of translation on a function?
A: The effect of translation on a function is a change in the value of the function. When a function is translated, it means that the original function is shifted or moved to a new position on the coordinate plane.
Q: What are the different types of translation?
A: There are two main types of translation: horizontal and vertical. Horizontal translation involves moving a function left or right on the coordinate plane, while vertical translation involves moving a function up or down on the coordinate plane.
Q: What is the effect of vertical translation on a function?
A: The effect of vertical translation on a function is a change in the value of the function. When a function is translated vertically, the value of the function changes, but the shape of the function remains the same.
Q: How do you determine the effect of vertical translation on a function?
A: To determine the effect of vertical translation on a function, you need to compare the value of the original function with the value of the translated function. The difference between the two values is the amount of the vertical translation.
Q: What is the effect of horizontal translation on a function?
A: The effect of horizontal translation on a function is a change in the position of the function on the coordinate plane. When a function is translated horizontally, the value of the function remains the same, but the position of the function changes.
Q: How do you determine the effect of horizontal translation on a function?
A: To determine the effect of horizontal translation on a function, you need to compare the position of the original function with the position of the translated function. The difference between the two positions is the amount of the horizontal translation.
Q: Can a function be translated both horizontally and vertically?
A: Yes, a function can be translated both horizontally and vertically. When a function is translated both horizontally and vertically, the effect is a combination of the two translations.
Q: How do you represent the effect of translation on a function graphically?
A: The effect of translation on a function can be represented graphically by shifting the original function to a new position on the coordinate plane. The amount of the translation is represented by the distance between the original function and the translated function.
Q: What are some real-world examples of translation?
A: Some real-world examples of translation include:
- Moving a object from one location to another
- Changing the value of a variable in a mathematical equation
- Shifting a graph to a new position on the coordinate plane
Conclusion
In conclusion, the effect of translation on a function is a change in the value or position of the function. Understanding the effect of translation is important in mathematics and real-world applications.
Key Takeaways
- The effect of translation on a function is a change in the value or position of the function.
- Vertical translation involves moving a function up or down on the coordinate plane.
- Horizontal translation involves moving a function left or right on the coordinate plane.
- A function can be translated both horizontally and vertically.
References
- [1] "Functions and Graphs" by Michael Sullivan
- [2] "Mathematics for the Nonmathematical" by Morris Kline
Discussion
What are some other examples of functions that can be translated? How do you think the effect of translation would change if the function was translated in a non-linear way?