Select The Correct Answer.Consider The Function $f(x) = 10^x$ And The Function $g(x$\], Which Is Shown Below. How Will The Graph Of $g(x$\] Differ From The Graph Of $f(x$\]?$g(x) = F(x-6) = 10^{(x-6)}$A. The

When dealing with functions, it's essential to understand how different transformations affect their graphical representation. In this article, we'll explore how the graph of a function $g(x)$ differs from the graph of a function $f(x)$ when $g(x)$ is a transformation of $f(x)$.

The Original Function: $f(x) = 10^x$

The function $f(x) = 10^x$ is an exponential function with a base of 10. This function has a characteristic "S" shape, where the value of the function increases rapidly as $x$ increases. The graph of this function is a continuous, smooth curve that passes through the point (0, 1).

The Transformed Function: $g(x) = f(x-6) = 10^{(x-6)}$

The function $g(x) = f(x-6) = 10^{(x-6)}$ is a transformation of the original function $f(x)$. This transformation involves shifting the graph of $f(x)$ to the right by 6 units. In other words, the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 6 units to the right.

How the Graph of $g(x)$ Differs from the Graph of $f(x)$

So, how does the graph of $g(x)$ differ from the graph of $f(x)$? The main difference is the horizontal shift of 6 units. The graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 6 units to the right. This means that for any given value of $x$, the value of $g(x)$ is the same as the value of $f(x-6)$.

Visualizing the Transformation

To visualize the transformation, let's consider a specific point on the graph of $f(x)$. Suppose we choose the point (2, 100). This point lies on the graph of $f(x)$ because $f(2) = 10^2 = 100$. Now, let's find the corresponding point on the graph of $g(x)$. Since the graph of $g(x)$ is shifted 6 units to the right, the point (2, 100) on the graph of $f(x)$ becomes the point (8, 100) on the graph of $g(x)$.

Key Takeaways

In conclusion, the graph of $g(x) = f(x-6) = 10^{(x-6)}$ differs from the graph of $f(x) = 10^x$ in that it is shifted 6 units to the right. This transformation involves shifting the graph of $f(x)$ to the right by 6 units, resulting in a new graph that is identical to the original graph but shifted horizontally.

Common Misconceptions

When dealing with transformations of functions, it's essential to understand the concept of horizontal shifts. A common misconception is that a horizontal shift involves moving the graph of a function up or down. However, a horizontal shift involves moving the graph of a function left or right.

Real-World Applications

Understanding the graphical representation of functions and their transformations has numerous real-world applications. For example, in physics, the motion of an object can be represented by a function that describes its position over time. By applying transformations to this function, we can model different types of motion, such as uniform motion or accelerated motion.

Conclusion

In conclusion, the graph of $g(x) = f(x-6) = 10^{(x-6)}$ differs from the graph of $f(x) = 10^x$ in that it is shifted 6 units to the right. This transformation involves shifting the graph of $f(x)$ to the right by 6 units, resulting in a new graph that is identical to the original graph but shifted horizontally. By understanding the graphical representation of functions and their transformations, we can model real-world phenomena and make predictions about the behavior of complex systems.

Frequently Asked Questions

Q: What is the difference between the graph of $g(x)$ and the graph of $f(x)$?

A: The graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 6 units to the right.

Q: How does the transformation $g(x) = f(x-6)$ affect the graph of $f(x)$?

A: The transformation $g(x) = f(x-6)$ shifts the graph of $f(x)$ to the right by 6 units.

Q: What is the significance of understanding the graphical representation of functions and their transformations?

In our previous article, we explored how the graph of a function $g(x)$ differs from the graph of a function $f(x)$ when $g(x)$ is a transformation of $f(x)$. In this article, we'll answer some frequently asked questions about the graphical representation of functions and their transformations.

Q: What is the difference between a horizontal shift and a vertical shift?

A: A horizontal shift involves moving the graph of a function left or right, while a vertical shift involves moving the graph of a function up or down.

Q: How do I determine the direction of a horizontal shift?

A: To determine the direction of a horizontal shift, you need to look at the sign of the value being added to or subtracted from the input variable $x$. If the value is positive, the graph will be shifted to the right. If the value is negative, the graph will be shifted to the left.

Q: What is the effect of a horizontal shift on the graph of a function?

A: A horizontal shift affects the graph of a function by changing the position of the graph along the x-axis. The graph is shifted to the right if the value being added to or subtracted from $x$ is positive, and to the left if the value is negative.

Q: How do I determine the effect of a vertical shift on the graph of a function?

A: To determine the effect of a vertical shift on the graph of a function, you need to look at the sign of the value being added to or subtracted from the function. If the value is positive, the graph will be shifted up. If the value is negative, the graph will be shifted down.

Q: What is the effect of a vertical shift on the graph of a function?

A: A vertical shift affects the graph of a function by changing the position of the graph along the y-axis. The graph is shifted up if the value being added to or subtracted from the function is positive, and down if the value is negative.

Q: How do I determine the effect of a combination of horizontal and vertical shifts on the graph of a function?

A: To determine the effect of a combination of horizontal and vertical shifts on the graph of a function, you need to look at the signs of the values being added to or subtracted from the input variable $x$ and the function. If the value being added to or subtracted from $x$ is positive, the graph will be shifted to the right. If the value is negative, the graph will be shifted to the left. If the value being added to or subtracted from the function is positive, the graph will be shifted up. If the value is negative, the graph will be shifted down.

Q: What is the significance of understanding the graphical representation of functions and their transformations?

A: Understanding the graphical representation of functions and their transformations has numerous real-world applications, including modeling motion and predicting the behavior of complex systems.

Q: How do I apply the concepts of horizontal and vertical shifts to real-world problems?

A: To apply the concepts of horizontal and vertical shifts to real-world problems, you need to identify the type of shift that is occurring and determine the effect of the shift on the graph of the function. This can be done by analyzing the problem and identifying the values being added to or subtracted from the input variable $x$ and the function.

Q: What are some common applications of the graphical representation of functions and their transformations?

A: Some common applications of the graphical representation of functions and their transformations include:

• Modeling motion: The graphical representation of functions and their transformations can be used to model the motion of objects, including the position, velocity, and acceleration of the object over time.
• Predicting the behavior of complex systems: The graphical representation of functions and their transformations can be used to predict the behavior of complex systems, including the behavior of populations, economies, and other complex systems.
• Analyzing data: The graphical representation of functions and their transformations can be used to analyze data, including the identification of trends and patterns in the data.

Conclusion

In conclusion, understanding the graphical representation of functions and their transformations is essential for a wide range of applications, including modeling motion and predicting the behavior of complex systems. By understanding the concepts of horizontal and vertical shifts, you can apply the graphical representation of functions and their transformations to real-world problems and make predictions about the behavior of complex systems.