Transformation of the Function

Oct 13, 2025 by talynt.org 23 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their transformations is crucial for solving various mathematical problems. In this article, we will explore the transformation of the function $f(x)=\log_5(x+7)$ from its parent function and identify the correct transformation among the given options.

Parent Function

The parent function of a logarithmic function is the basic function without any transformations. In this case, the parent function is $f(x)=\log_5x$ . This function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=-\infty$ .

Transformation of the Function

The given function is $f(x)=\log_5(x+7)$ . To identify the transformation, we need to compare it with the parent function. The key difference between the two functions is the presence of the term $(x+7)$ inside the logarithm.

Vertical Shift

A vertical shift occurs when the function is shifted up or down by a certain value. In this case, the function is shifted up by 7 units. To see this, we can rewrite the function as $f(x)=\log_5(x+7)-\log_55$ . Using the properties of logarithms, we can simplify this expression to $f(x)=\log_5\frac{x+7}{5}$ .

Horizontal Shift

A horizontal shift occurs when the function is shifted left or right by a certain value. In this case, the function is shifted left by 7 units. To see this, we can rewrite the function as $f(x-7)=\log_5(x-7+7)$ . This simplifies to $f(x-7)=\log_5x$ .

Conclusion

Based on the analysis above, we can conclude that the correct transformation of the function $f(x)=\log_5(x+7)$ from its parent function is a vertical shift up 7 units.

Answer

The correct answer is A. Vertical shift up 7 units.

Additional Information

The horizontal shift left 7 units (option B) is incorrect because the function is shifted left by 7 units, not right.
The horizontal shift right 7 units (option C) is also incorrect because the function is shifted left by 7 units, not right.
The vertical shift down 7 units (option D) is incorrect because the function is shifted up by 7 units, not down.

Real-World Applications

Understanding the transformation of logarithmic functions has various real-world applications, such as:

Modeling population growth and decay
Analyzing financial data and predicting stock prices
Studying the behavior of chemical reactions and physical systems

Tips and Tricks

When analyzing the transformation of a function, start by identifying the parent function.
Compare the given function with the parent function to identify the transformation.
Use the properties of logarithms to simplify the expression and identify the transformation.

Conclusion

Q: What is the parent function of a logarithmic function?

A: The parent function of a logarithmic function is the basic function without any transformations. For example, the parent function of $f(x)=\log_5(x+7)$ is $f(x)=\log_5x$ .

Q: How do I identify the transformation of a logarithmic function?

A: To identify the transformation of a logarithmic function, compare the given function with its parent function. Look for any changes in the argument of the logarithm, such as a horizontal shift or a vertical shift.

Q: What is a vertical shift in a logarithmic function?

A: A vertical shift in a logarithmic function occurs when the function is shifted up or down by a certain value. For example, the function $f(x)=\log_5(x+7)$ is shifted up by 7 units compared to its parent function $f(x)=\log_5x$ .

Q: What is a horizontal shift in a logarithmic function?

A: A horizontal shift in a logarithmic function occurs when the function is shifted left or right by a certain value. For example, the function $f(x)=\log_5(x-7)$ is shifted left by 7 units compared to its parent function $f(x)=\log_5x$ .

Q: How do I determine the type of shift in a logarithmic function?

A: To determine the type of shift in a logarithmic function, look at the argument of the logarithm. If the argument is increased by a constant, it is a vertical shift. If the argument is decreased by a constant, it is a horizontal shift.

Q: Can a logarithmic function have both a vertical and horizontal shift?

A: Yes, a logarithmic function can have both a vertical and horizontal shift. For example, the function $f(x)=\log_5(x-7+7)$ has both a vertical and horizontal shift compared to its parent function $f(x)=\log_5x$ .

Q: How do I apply the properties of logarithms to simplify a logarithmic function?

A: To apply the properties of logarithms to simplify a logarithmic function, use the following rules:

$\log_a(b+c)=\log_a(b)+\log_a(c+1)$
$\log_a(bc)=\log_a(b)+\log_a(c)$
$\log_a\frac{b}{c}=\log_a(b)-\log_a(c)$

Q: What are some real-world applications of logarithmic function transformations?

A: Logarithmic function transformations have various real-world applications, such as:

Modeling population growth and decay
Analyzing financial data and predicting stock prices
Studying the behavior of chemical reactions and physical systems

Q: How can I use logarithmic function transformations to solve problems?

A: To use logarithmic function transformations to solve problems, follow these steps:

Identify the parent function of the given logarithmic function.
Compare the given function with its parent function to identify the transformation.
Use the properties of logarithms to simplify the expression and identify the transformation.
Apply the transformation to the parent function to obtain the solution.

Conclusion

In conclusion, logarithmic function transformations are a crucial concept in mathematics, and understanding them is essential for solving various mathematical problems. By following the tips and tricks provided above, you can analyze the transformation of logarithmic functions with ease and apply them to real-world problems.