Consider The Function Y = 15 X Y = 15^x Y = 1 5 X . How Do The Y Y Y -values Of This Function Grow?A. By Adding 15B. By Multiplying The Previous Y Y Y -value By 15C. By Adding 15, Then 30, Then 45, And So On

Mar 12, 2025 by ADMIN 208 views

Understanding the Growth of y-Values in the Function y = 15^x

The function $y = 15^x$ is an exponential function that exhibits rapid growth as the value of $x$ increases. In this discussion, we will explore how the $y$ -values of this function grow, and examine the characteristics of exponential growth.

Exponential growth occurs when a quantity increases by a fixed percentage or factor at regular intervals. In the case of the function $y = 15^x$ , the $y$ -values grow by a factor of 15 for each increase in $x$ by 1. This means that if the $y$ -value is 15 when $x$ is 1, it will be 225 when $x$ is 2, and 3375 when $x$ is 3.

How y-Values Grow

To understand how the $y$ -values grow, let's examine the function $y = 15^x$ more closely. When $x$ is 1, $y$ is equal to 15. When $x$ is 2, $y$ is equal to 15^2, which is 225. When $x$ is 3, $y$ is equal to 15^3, which is 3375. As we can see, the $y$ -values are growing by a factor of 15 for each increase in $x$ by 1.

Multiplying the Previous y-Value by 15

The key characteristic of exponential growth is that each new value is obtained by multiplying the previous value by a fixed factor. In the case of the function $y = 15^x$ , the fixed factor is 15. This means that to find the next value of $y$ , we simply multiply the previous value of $y$ by 15.

Example

Let's consider an example to illustrate how the $y$ -values grow. Suppose we want to find the value of $y$ when $x$ is 4. We can start with the value of $y$ when $x$ is 3, which is 3375. To find the value of $y$ when $x$ is 4, we multiply 3375 by 15, which gives us 50625.

In conclusion, the $y$ -values of the function $y = 15^x$ grow by multiplying the previous value by 15 for each increase in $x$ by 1. This is a classic example of exponential growth, where each new value is obtained by multiplying the previous value by a fixed factor. Understanding how the $y$ -values grow is essential for working with exponential functions and making predictions about their behavior.

The function $y = 15^x$ is an exponential function that exhibits rapid growth as the value of $x$ increases.
The $y$ -values grow by a factor of 15 for each increase in $x$ by 1.
Exponential growth occurs when a quantity increases by a fixed percentage or factor at regular intervals.
Each new value is obtained by multiplying the previous value by a fixed factor.

For more information on exponential functions and their applications, see the following resources:

Calculus
Algebra
Geometry
Q&A: Understanding the Function y = 15^x

In our previous discussion, we explored the function $y = 15^x$ and how the $y$ -values grow. In this Q&A article, we will answer some common questions about the function and its behavior.

Q: What is the function y = 15^x?

A: The function $y = 15^x$ is an exponential function that exhibits rapid growth as the value of $x$ increases. The function can be written as $y = 15^x$ , where $x$ is the input and $y$ is the output.

Q: How do the y-values grow in the function y = 15^x?

A: The $y$ -values grow by a factor of 15 for each increase in $x$ by 1. This means that if the $y$ -value is 15 when $x$ is 1, it will be 225 when $x$ is 2, and 3375 when $x$ is 3.

Q: What is the key characteristic of exponential growth?

A: The key characteristic of exponential growth is that each new value is obtained by multiplying the previous value by a fixed factor. In the case of the function $y = 15^x$ , the fixed factor is 15.

Q: How can I calculate the value of y when x is a specific value?

A: To calculate the value of $y$ when $x$ is a specific value, you can use the formula $y = 15^x$ . For example, if you want to find the value of $y$ when $x$ is 4, you can multiply the value of $y$ when $x$ is 3 by 15.

Q: What are some real-world applications of the function y = 15^x?

A: The function $y = 15^x$ has many real-world applications, including:

Modeling population growth
Calculating compound interest
Analyzing the growth of investments
Understanding the behavior of exponential functions in various fields

Q: Can I use the function y = 15^x to model other types of growth?

A: Yes, you can use the function $y = 15^x$ to model other types of growth, such as population growth, chemical reactions, and financial investments. However, you may need to adjust the base value (15) and the exponent (x) to fit the specific problem you are trying to model.

Q: How can I graph the function y = 15^x?

A: You can graph the function $y = 15^x$ using a graphing calculator or a computer program. The graph will show an exponential curve that grows rapidly as the value of $x$ increases.

Q: What are some common mistakes to avoid when working with the function y = 15^x?

A: Some common mistakes to avoid when working with the function $y = 15^x$ include:

Confusing the base value (15) with the exponent (x)
Failing to account for the fixed factor (15) in exponential growth
Not using the correct formula ( $y = 15^x$ ) to calculate the value of $y$

In conclusion, the function $y = 15^x$ is a powerful tool for modeling exponential growth and understanding the behavior of exponential functions. By understanding how the $y$ -values grow and using the correct formula, you can apply this function to a wide range of real-world problems.

The function $y = 15^x$ is an exponential function that exhibits rapid growth as the value of $x$ increases.
The $y$ -values grow by a factor of 15 for each increase in $x$ by 1.
Exponential growth occurs when a quantity increases by a fixed percentage or factor at regular intervals.
Each new value is obtained by multiplying the previous value by a fixed factor.

For more information on exponential functions and their applications, see the following resources: