Which Is The Graph Of $y=-\log (x-2)+3$?

Feb 26, 2025 by ADMIN 43 views

$**Understanding the Graph of $y=-\log (x-2)+3$**$

Introduction to Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. A logarithmic function is the inverse of an exponential function, and it is used to represent the power or exponent to which a base number must be raised to produce a given value. In this article, we will focus on the graph of the logarithmic function $y=-\log (x-2)+3$.

The Basics of Logarithmic Graphs

A logarithmic graph is a type of graph that represents a logarithmic function. The graph of a logarithmic function is a curve that increases or decreases as the input value increases or decreases. The graph of a logarithmic function has several key features, including:

Vertical asymptote: A vertical asymptote is a vertical line that the graph approaches but never touches. In the case of a logarithmic function, the vertical asymptote is at the value of x that makes the argument of the logarithm equal to zero.
Horizontal asymptote: A horizontal asymptote is a horizontal line that the graph approaches as x approaches infinity or negative infinity. In the case of a logarithmic function, the horizontal asymptote is at the value of y that is equal to the logarithm of the base.
Intercepts: The x-intercept is the point where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.

Graphing the Function $y=-\log (x-2)+3$

To graph the function $y=-\log (x-2)+3$, we need to consider the following:

Vertical asymptote: The vertical asymptote is at x = 2, since the argument of the logarithm is x - 2.
Horizontal asymptote: The horizontal asymptote is at y = 3, since the function is of the form y = -log(x - 2) + 3.
Intercepts: The x-intercept is at x = 2, since the function is equal to zero at this point. The y-intercept is at y = 3, since the function is equal to 3 at this point.

Key Features of the Graph

The graph of the function $y=-\log (x-2)+3$ has several key features, including:

Vertical asymptote: The graph has a vertical asymptote at x = 2.
Horizontal asymptote: The graph has a horizontal asymptote at y = 3.
Intercepts: The graph has an x-intercept at x = 2 and a y-intercept at y = 3.
Increasing or decreasing: The graph is decreasing as x increases, since the function is of the form y = -log(x - 2) + 3.

Real-World Applications

The graph of the function $y=-\log (x-2)+3$ has several real-world applications, including:

Physics: The graph can be used to model the behavior of a physical system that exhibits logarithmic behavior.
Engineering: The graph can be used to design and optimize systems that involve logarithmic functions.
Economics: The graph can be used to model the behavior of economic systems that involve logarithmic functions.

Conclusion

In conclusion, the graph of the function $y=-\log (x-2)+3$ is a logarithmic graph that has several key features, including a vertical asymptote, a horizontal asymptote, and intercepts. The graph is decreasing as x increases, and it has several real-world applications in physics, engineering, and economics.

References

[1] Logarithmic Functions. In: Mathematics for Engineers and Scientists. Springer, Cham.
[2] Graphing Logarithmic Functions. In: Mathematics for Engineers and Scientists. Springer, Cham.
[3] Real-World Applications of Logarithmic Functions. In: Mathematics for Engineers and Scientists. Springer, Cham.

Glossary

Logarithmic function: A function that represents the power or exponent to which a base number must be raised to produce a given value.
Vertical asymptote: A vertical line that the graph approaches but never touches.
Horizontal asymptote: A horizontal line that the graph approaches as x approaches infinity or negative infinity.
Intercepts: The points where the graph crosses the x-axis and the y-axis.

FAQs

What is the vertical asymptote of the graph of the function $y=-\log (x-2)+3$?
- The vertical asymptote is at x = 2.
What is the horizontal asymptote of the graph of the function $y=-\log (x-2)+3$?
- The horizontal asymptote is at y = 3.
What are the intercepts of the graph of the function $y=-\log (x-2)+3$?
- The x-intercept is at x = 2, and the y-intercept is at y = 3.

Q: What is the vertical asymptote of the graph of the function $y=-\log (x-2)+3$?

A: The vertical asymptote is at x = 2. This is because the argument of the logarithm is x - 2, and the logarithm is undefined when the argument is equal to zero.

Q: What is the horizontal asymptote of the graph of the function $y=-\log (x-2)+3$?

A: The horizontal asymptote is at y = 3. This is because the function is of the form y = -log(x - 2) + 3, and the constant term 3 is the horizontal asymptote.

Q: What are the intercepts of the graph of the function $y=-\log (x-2)+3$?

A: The x-intercept is at x = 2, and the y-intercept is at y = 3. This is because the function is equal to zero at x = 2, and the function is equal to 3 at y = 3.

Q: Is the graph of the function $y=-\log (x-2)+3$ increasing or decreasing?

A: The graph is decreasing as x increases. This is because the function is of the form y = -log(x - 2) + 3, and the negative sign in front of the logarithm causes the function to decrease as x increases.

Q: What is the domain of the function $y=-\log (x-2)+3$?

A: The domain of the function is all real numbers except x = 2. This is because the argument of the logarithm is x - 2, and the logarithm is undefined when the argument is equal to zero.

Q: What is the range of the function $y=-\log (x-2)+3$?

A: The range of the function is all real numbers except y = 3. This is because the function is of the form y = -log(x - 2) + 3, and the constant term 3 is the range.

Q: How can I graph the function $y=-\log (x-2)+3$?

A: You can graph the function by using a graphing calculator or by plotting points on a coordinate plane. To plot points, you can use the following steps:

Choose a value of x.
Calculate the corresponding value of y using the function y = -log(x - 2) + 3.
Plot the point (x, y) on a coordinate plane.

Q: What are some real-world applications of the graph of the function $y=-\log (x-2)+3$?

A: Some real-world applications of the graph of the function include:

Physics: The graph can be used to model the behavior of a physical system that exhibits logarithmic behavior.
Engineering: The graph can be used to design and optimize systems that involve logarithmic functions.
Economics: The graph can be used to model the behavior of economic systems that involve logarithmic functions.

Q: How can I use the graph of the function $y=-\log (x-2)+3$ to solve problems?

A: You can use the graph to solve problems by using the following steps:

Identify the vertical asymptote and horizontal asymptote of the graph.
Identify the intercepts of the graph.
Use the graph to determine the behavior of the function as x increases or decreases.
Use the graph to solve problems that involve logarithmic functions.

Q: What are some common mistakes to avoid when graphing the function $y=-\log (x-2)+3$?

A: Some common mistakes to avoid when graphing the function include:

Not identifying the vertical asymptote and horizontal asymptote: Make sure to identify the vertical asymptote and horizontal asymptote of the graph.
Not identifying the intercepts: Make sure to identify the intercepts of the graph.
Not using the correct scale: Make sure to use the correct scale when graphing the function.
Not plotting enough points: Make sure to plot enough points to get an accurate graph.

Q: How can I check my work when graphing the function $y=-\log (x-2)+3$?

A: You can check your work by using the following steps:

Verify that the vertical asymptote and horizontal asymptote are correct.
Verify that the intercepts are correct.
Verify that the graph is accurate and complete.
Use a graphing calculator or software to check your work.

Q: What are some additional resources for learning about the graph of the function $y=-\log (x-2)+3$?

A: Some additional resources for learning about the graph of the function include:

Textbooks: There are many textbooks that cover logarithmic functions and their graphs.
Online resources: There are many online resources that provide information and examples about logarithmic functions and their graphs.
Software: There are many software programs that can be used to graph logarithmic functions and their graphs.
Tutorials: There are many tutorials that provide step-by-step instructions for graphing logarithmic functions and their graphs.