Describe Which Way (up/down, Left/right) And By How Much The Graph Of $f(x) = \log X$ Will Shift When Its Parameters Are Changed To $f(x) = \log (x + 5)$.Which Way? □ \square □ How Much? □ \square □ Options: A. 5

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding how they shift when their parameters are changed is crucial for solving various mathematical problems. In this article, we will explore how the graph of the function $f(x) = \log x$ shifts when its parameters are changed to $f(x) = \log (x + 5)$.

The Original Function

The original function is $f(x) = \log x$. This function is a logarithmic function with a base of 10, and it is defined for all positive real numbers $x$. The graph of this function is a curve that increases as $x$ increases.

The Modified Function

The modified function is $f(x) = \log (x + 5)$. This function is also a logarithmic function with a base of 10, but it is defined for all positive real numbers $x$ such that $x + 5 > 0$. The graph of this function is also a curve that increases as $x$ increases.

Shifting the Graph

When the parameters of the function are changed from $f(x) = \log x$ to $f(x) = \log (x + 5)$, the graph of the function shifts to the left by 5 units. This is because the argument of the logarithmic function is now $x + 5$, which is 5 units to the left of $x$.

Why the Shift Occurs

The shift occurs because the argument of the logarithmic function is now $x + 5$, which is 5 units to the left of $x$. This means that for any given value of $x$, the corresponding value of $x + 5$ is 5 units to the left of $x$. As a result, the graph of the function shifts to the left by 5 units.

How Much the Graph Shifts

The graph shifts by 5 units to the left. This means that for any given value of $x$, the corresponding value of $x + 5$ is 5 units to the left of $x$.

Conclusion

In conclusion, when the parameters of the function $f(x) = \log x$ are changed to $f(x) = \log (x + 5)$, the graph of the function shifts to the left by 5 units. This shift occurs because the argument of the logarithmic function is now $x + 5$, which is 5 units to the left of $x$.

Key Takeaways

• The graph of the function $f(x) = \log x$ shifts to the left by 5 units when its parameters are changed to $f(x) = \log (x + 5)$.
• The shift occurs because the argument of the logarithmic function is now $x + 5$, which is 5 units to the left of $x$.
• The graph shifts by 5 units to the left.

Final Answer

Introduction

In our previous article, we explored how the graph of the function $f(x) = \log x$ shifts when its parameters are changed to $f(x) = \log (x + 5)$. In this article, we will answer some frequently asked questions about the shift in logarithmic functions.

Q: What is the direction of the shift?

A: The graph of the function $f(x) = \log x$ shifts to the left by 5 units when its parameters are changed to $f(x) = \log (x + 5)$.

Q: Why does the graph shift to the left?

A: The graph shifts to the left because the argument of the logarithmic function is now $x + 5$, which is 5 units to the left of $x$.

Q: How much does the graph shift?

A: The graph shifts by 5 units to the left.

Q: What happens to the graph if the parameter is changed to $f(x) = \log (x - 5)$?

A: If the parameter is changed to $f(x) = \log (x - 5)$, the graph of the function will shift to the right by 5 units.

Q: Why does the graph shift to the right in this case?

A: The graph shifts to the right because the argument of the logarithmic function is now $x - 5$, which is 5 units to the right of $x$.

Q: Can the graph shift in both directions?

A: Yes, the graph can shift in both directions depending on the change in the parameter. If the parameter is changed to $f(x) = \log (x + 5)$, the graph shifts to the left. If the parameter is changed to $f(x) = \log (x - 5)$, the graph shifts to the right.

Q: How does the shift affect the graph's asymptote?

A: The shift does not affect the graph's asymptote. The asymptote remains the same, but the graph is shifted horizontally.

Q: Can the shift be applied to other types of functions?

A: Yes, the shift can be applied to other types of functions, not just logarithmic functions. However, the direction and amount of the shift will depend on the specific function and the change in its parameter.

Q: What is the significance of the shift in logarithmic functions?

A: The shift in logarithmic functions is significant because it allows us to model real-world phenomena that involve exponential growth or decay. By understanding how the graph shifts, we can better analyze and predict the behavior of these phenomena.

Conclusion

In conclusion, the shift in logarithmic functions is an important concept that can help us better understand and analyze real-world phenomena. By answering some frequently asked questions about the shift, we hope to have provided a clearer understanding of this concept.

Key Takeaways

• The graph of the function $f(x) = \log x$ shifts to the left by 5 units when its parameters are changed to $f(x) = \log (x + 5)$.
• The graph shifts to the right by 5 units when its parameters are changed to $f(x) = \log (x - 5)$.
• The shift can be applied to other types of functions, but the direction and amount of the shift will depend on the specific function and the change in its parameter.
• The shift is significant because it allows us to model real-world phenomena that involve exponential growth or decay.