The Exponential Function $p(x)$ Increases At A Rate Of $25\%$ Through The Ordered Pair \$(0,10)$[/tex\] And Is Shifted Down 5 Units. Use A Graph Of The Function To Determine The Range.A. $(5, \infty$\] B.

Introduction

The exponential function is a fundamental concept in mathematics, and it has numerous applications in various fields, including finance, science, and engineering. In this article, we will explore the exponential function $p(x)$, which increases at a rate of $25\%$ through the ordered pair $(0,10)$ and is shifted down 5 units. We will use a graph of the function to determine its range.

Understanding the Exponential Function

The exponential function is a mathematical function that is defined as $f(x) = a^x$, where $a$ is a positive real number. The function $p(x)$ is an example of an exponential function, where $a = 1.25$ (since it increases at a rate of $25\%$). The function can be written as $p(x) = 1.25^x$.

Graphing the Function

To graph the function $p(x)$, we can start by plotting the point $(0,10)$, which is given in the problem statement. Since the function increases at a rate of $25\%$, we can plot the point $(1,12.5)$, which is one unit to the right of the point $(0,10)$. We can continue this process to plot more points on the graph.

Shifting the Function Down 5 Units

The function $p(x)$ is shifted down 5 units, which means that we need to subtract 5 from the y-coordinate of each point on the graph. This will result in a new graph that is 5 units below the original graph.

Determining the Range

To determine the range of the function $p(x)$, we need to find the set of all possible y-values that the function can take. Since the function is an exponential function, it will continue to increase without bound as $x$ increases. Therefore, the range of the function will be all real numbers greater than or equal to the minimum value of the function.

Finding the Minimum Value

To find the minimum value of the function, we can use the fact that the function is shifted down 5 units. This means that the minimum value of the function will be 5 units below the minimum value of the original function. Since the original function has a minimum value of 10, the minimum value of the shifted function will be $10 - 5 = 5$.

Conclusion

In conclusion, the range of the function $p(x)$ is all real numbers greater than or equal to 5. This can be written as $(5, \infty)$.

Final Answer

The final answer is $\boxed{(5, \infty)}$.

Discussion

The exponential function is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we have explored the exponential function $p(x)$, which increases at a rate of $25\%$ through the ordered pair $(0,10)$ and is shifted down 5 units. We have used a graph of the function to determine its range, which is all real numbers greater than or equal to 5.

Related Topics

• Exponential functions
• Graphing functions
• Shifting functions
• Determining the range of a function

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Graphing Functions" by Khan Academy
• [3] "Shifting Functions" by Purplemath
• [4] "Determining the Range of a Function" by Math Is Fun
  

Introduction

In our previous article, we explored the exponential function $p(x)$, which increases at a rate of $25\%$ through the ordered pair $(0,10)$ and is shifted down 5 units. We used a graph of the function to determine its range, which is all real numbers greater than or equal to 5. In this article, we will answer some frequently asked questions about the exponential function and its range.

Q: What is the exponential function $p(x)$?

A: The exponential function $p(x)$ is a mathematical function that is defined as $p(x) = 1.25^x$, where $x$ is a real number. The function increases at a rate of $25\%$ through the ordered pair $(0,10)$ and is shifted down 5 units.

Q: How do I graph the function $p(x)$?

A: To graph the function $p(x)$, you can start by plotting the point $(0,10)$, which is given in the problem statement. Since the function increases at a rate of $25\%$, you can plot the point $(1,12.5)$, which is one unit to the right of the point $(0,10)$. You can continue this process to plot more points on the graph.

Q: What is the range of the function $p(x)$?

A: The range of the function $p(x)$ is all real numbers greater than or equal to 5. This can be written as $(5, \infty)$.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to find the set of all possible y-values that the function can take. You can use the graph of the function to help you determine the range.

Q: What is the minimum value of the function $p(x)$?

A: The minimum value of the function $p(x)$ is 5, which is 5 units below the minimum value of the original function.

Q: Can I use the exponential function $p(x)$ in real-world applications?

A: Yes, the exponential function $p(x)$ has numerous applications in various fields, including finance, science, and engineering. For example, you can use the function to model population growth, chemical reactions, and financial investments.

Q: How do I 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 y-coordinate of each point on the graph.

Q: Can I use the graphing method to determine the range of any function?

A: Yes, you can use the graphing method to determine the range of any function. However, you need to make sure that the graph is accurate and complete.

Q: What are some common mistakes to avoid when graphing a function?

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

• Not plotting enough points on the graph
• Not using a ruler or other straightedge to draw the graph
• Not labeling the axes and the graph
• Not checking the accuracy of the graph

Conclusion

In conclusion, the exponential function $p(x)$ is a fundamental concept in mathematics, and it has numerous applications in various fields. We have answered some frequently asked questions about the exponential function and its range, and we hope that this article has been helpful to you.

Final Answer

The final answer is $\boxed{(5, \infty)}$.

Discussion

The exponential function is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we have answered some frequently asked questions about the exponential function and its range. We hope that this article has been helpful to you.

Related Topics

• Exponential functions
• Graphing functions
• Shifting functions
• Determining the range of a function

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Graphing Functions" by Khan Academy
• [3] "Shifting Functions" by Purplemath
• [4] "Determining the Range of a Function" by Math Is Fun