Shrinking the Graph Vertically

Oct 13, 2025 by miodeya.com 13 views

Introduction

In mathematics, functions are used to describe the relationship between two variables. When working with functions, it's often necessary to perform transformations on the graph of the function. These transformations can include shifting the graph up or down, shrinking or expanding the graph vertically or horizontally, and reflecting the graph across the x-axis or y-axis. In this article, we will explore how to construct a new function $g(x)$ that is equivalent to the original function $f(x) = \log_b(x)$ but has been shifted 7 units down, shrunk vertically by a factor of 3, and reflected across the x-axis.

Shifting the Graph Down

To shift the graph of $f(x)$ down by 7 units, we need to subtract 7 from the function. This can be achieved by adding a constant term to the function. The new function $g(x)$ will be of the form:

g(x) = f(x) - 7

Since $f(x) = \log_b(x)$ , we can substitute this into the equation for $g(x)$ :

g(x) = \log_b(x) - 7

Shrinking the Graph Vertically

To shrink the graph of $g(x)$ vertically by a factor of 3, we need to multiply the function by $\frac{1}{3}$ . This can be achieved by multiplying the function by a constant term. The new function $g(x)$ will be of the form:

g(x) = \frac{1}{3} \cdot g(x)

Substituting the equation for $g(x)$ into this equation, we get:

g(x) = \frac{1}{3} \cdot (\log_b(x) - 7)

Reflecting the Graph Across the X-Axis

To reflect the graph of $g(x)$ across the x-axis, we need to multiply the function by -1. This can be achieved by multiplying the function by a constant term. The new function $g(x)$ will be of the form:

g(x) = -1 \cdot g(x)

Substituting the equation for $g(x)$ into this equation, we get:

g(x) = -1 \cdot \left(\frac{1}{3} \cdot (\log_b(x) - 7)\right)

Simplifying this equation, we get:

g(x) = -\frac{1}{3} \cdot (\log_b(x) - 7)

g(x) = -\frac{1}{3} \log_b(x) + \frac{7}{3}

The Final Function

The final function $g(x)$ that is equivalent to the original function $f(x) = \log_b(x)$ but has been shifted 7 units down, shrunk vertically by a factor of 3, and reflected across the x-axis is:

g(x) = -\frac{1}{3} \log_b(x) + \frac{7}{3}

Example

Let's consider an example to illustrate how to use the function $g(x)$ . Suppose we want to find the value of $g(10)$ .

Substituting $x = 10$ into the equation for $g(x)$ , we get:

g(10) = -\frac{1}{3} \log_b(10) + \frac{7}{3}

Using a calculator to evaluate the logarithm, we get:

\log_b(10) \approx 1.04139

Substituting this value into the equation for $g(10)$ , we get:

g(10) = -\frac{1}{3} \cdot 1.04139 + \frac{7}{3}

Simplifying this equation, we get:

g(10) \approx -0.34713 + 2.33333

g(10) \approx 2.0

Therefore, the value of $g(10)$ is approximately 2.0.

Conclusion

In this article, we have explored how to construct a new function $g(x)$ that is equivalent to the original function $f(x) = \log_b(x)$ but has been shifted 7 units down, shrunk vertically by a factor of 3, and reflected across the x-axis. We have derived the equation for $g(x)$ and provided an example to illustrate how to use the function. The final function $g(x)$ is:

g(x) = -\frac{1}{3} \log_b(x) + \frac{7}{3}

Q: What is the purpose of transforming functions?

A: The purpose of transforming functions is to change the graph of a function in various ways, such as shifting, shrinking, or reflecting. This can be useful in modeling real-world phenomena, such as population growth, chemical reactions, or electrical circuits.

Q: How do you shift a function down by a certain number of units?

A: To shift a function down by a certain number of units, you need to subtract that number from the function. For example, if you want to shift the function $f(x) = \log_b(x)$ down by 7 units, you would use the function $g(x) = f(x) - 7$ .

Q: How do you shrink a function vertically by a certain factor?

A: To shrink a function vertically by a certain factor, you need to multiply the function by that factor. For example, if you want to shrink the function $g(x) = \log_b(x) - 7$ vertically by a factor of 3, you would use the function $h(x) = \frac{1}{3} \cdot g(x)$ .

Q: How do you reflect a function across the x-axis?

A: To reflect a function across the x-axis, you need to multiply the function by -1. For example, if you want to reflect the function $h(x) = \frac{1}{3} \cdot (\log_b(x) - 7)$ across the x-axis, you would use the function $k(x) = -1 \cdot h(x)$ .

Q: What is the final function $g(x)$ that is equivalent to the original function $f(x) = \log_b(x)$ but has been shifted 7 units down, shrunk vertically by a factor of 3, and reflected across the x-axis?

A: The final function $g(x)$ is:

g(x) = -\frac{1}{3} \log_b(x) + \frac{7}{3}

Q: How do you use the function $g(x)$ to model real-world phenomena?

A: You can use the function $g(x)$ to model real-world phenomena by substituting the values of the variables into the equation and solving for the unknown quantity. For example, if you want to model the population growth of a certain species, you would substitute the values of the population size and the growth rate into the equation and solve for the population size at a future time.

Q: What are some examples of real-world phenomena that can be modeled using the function $g(x)$ ?

A: Some examples of real-world phenomena that can be modeled using the function $g(x)$ include:

Population growth
Chemical reactions
Electrical circuits
Financial models
Medical models

Q: How do you evaluate the function $g(x)$ at a certain value of x?

A: To evaluate the function $g(x)$ at a certain value of x, you need to substitute the value of x into the equation and solve for the value of the function. For example, if you want to evaluate the function $g(x) = -\frac{1}{3} \log_b(x) + \frac{7}{3}$ at x = 10, you would substitute x = 10 into the equation and solve for the value of the function.

Q: What are some common mistakes to avoid when working with functions?

A: Some common mistakes to avoid when working with functions include:

Not following the order of operations
Not using parentheses correctly
Not simplifying the function
Not checking the domain and range of the function

Q: How do you check the domain and range of a function?

A: To check the domain and range of a function, you need to examine the equation and determine the values of x and y that make the function undefined or undefined. For example, if you have a function $f(x) = \frac{1}{x}$ , the domain of the function is all real numbers except 0, and the range of the function is all real numbers except 0.

Q: What are some common applications of functions in real-world scenarios?

A: Some common applications of functions in real-world scenarios include:

Modeling population growth
Modeling chemical reactions
Modeling electrical circuits
Modeling financial models
Modeling medical models

Q: How do you use functions to solve real-world problems?

A: To use functions to solve real-world problems, you need to:

Identify the problem and the variables involved
Determine the type of function that is needed to model the problem
Write the equation for the function
Solve the equation for the unknown quantity
Interpret the results in the context of the problem

By following these steps, you can use functions to solve a wide range of real-world problems.