Determine Whether The Function Represents Exponential Growth Or Decay. Write The Base In Terms Of The Rate Of Growth Or Decay, Identify { T $} , A N D I N T E R P R E T T H E R A T E O F G R O W T H O R D E C A Y . T H E F U N C T I O N \[ , And Interpret The Rate Of Growth Or Decay.The Function \[ , An D In T Er P Re Tt H Er A T Eo F G Ro Wt H Or D Ec A Y . T H E F U N C T I O N \[ Y = 11,700 \left( \frac{7}{10}

Understanding Exponential Growth and Decay

Exponential growth and decay are two fundamental concepts in mathematics that describe how quantities change over time. Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in rapid growth. On the other hand, exponential decay occurs when a quantity decreases at a rate proportional to its current value, resulting in a gradual decrease.

Identifying Exponential Growth or Decay

To determine whether a function represents exponential growth or decay, we need to examine its general form. An exponential growth function typically has the form:

$$y = ab^x$$

where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time variable. If $b > 1$, the function represents exponential growth. If $0 < b < 1$, the function represents exponential decay.

Analyzing the Given Function

The given function is:

y = 11,700 \left( \frac{7}{10}

To determine whether this function represents exponential growth or decay, we need to examine its general form. The function can be rewritten as:

y = 11,700 \left( \frac{7}{10}

Comparing this with the general form of an exponential function, we can see that the growth factor is $\frac{7}{10}$. Since $0 < \frac{7}{10} < 1$, the function represents exponential decay.

Writing the Base in Terms of the Rate of Growth or Decay

The base of the exponential function is $\frac{7}{10}$. This represents the rate of decay, which is $10\%$ per time period.

Identifying { t $}$

The time variable $t$ is not explicitly given in the function. However, we can assume that $t$ represents the time period over which the decay occurs.

Interpreting the Rate of Growth or Decay

The rate of decay is $10\%$ per time period. This means that the quantity decreases by $10\%$ of its current value every time period. For example, if the initial value is $11,700$, after one time period, the value will be $10,430$. After two time periods, the value will be $9,281.30$, and so on.

Conclusion

In conclusion, the given function represents exponential decay. The base of the function is $\frac{7}{10}$, which represents the rate of decay. The time variable $t$ is not explicitly given, but we can assume that it represents the time period over which the decay occurs. The rate of decay is $10\%$ per time period, which means that the quantity decreases by $10\%$ of its current value every time period.

Exponential Growth and Decay: Key Concepts

• Exponential growth occurs when a quantity increases at a rate proportional to its current value.
• Exponential decay occurs when a quantity decreases at a rate proportional to its current value.
• The general form of an exponential function is $y = ab^x$, where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time variable.
• If $b > 1$, the function represents exponential growth. If $0 < b < 1$, the function represents exponential decay.

Examples of Exponential Growth and Decay

• Population growth: The population of a city grows exponentially, with a growth factor of $1.05$ per year.
• Radioactive decay: The amount of a radioactive substance decreases exponentially, with a decay factor of $0.95$ per year.
• Compound interest: The amount of money in a savings account grows exponentially, with a growth factor of $1.05$ per year.

Real-World Applications of Exponential Growth and Decay

• Population growth and decline: Exponential growth and decay are used to model population growth and decline in cities and countries.
• Radioactive decay: Exponential decay is used to model the decay of radioactive substances in nuclear reactors and medicine.
• Compound interest: Exponential growth is used to model the growth of money in savings accounts and investments.
• Epidemiology: Exponential growth and decay are used to model the spread of diseases and the effectiveness of treatments.

Solving Exponential Growth and Decay Problems

• To solve an exponential growth or decay problem, we need to identify the initial value, the growth or decay factor, and the time variable.
• We can use the general form of an exponential function to solve the problem.
• We can also use logarithms to solve exponential growth and decay problems.

Conclusion

$$$$$$$$$$$$

Q: What is exponential growth?

A: Exponential growth is a process where a quantity increases at a rate proportional to its current value. This means that the rate of growth is not constant, but rather it increases as the quantity grows.

Q: What is exponential decay?

A: Exponential decay is a process where a quantity decreases at a rate proportional to its current value. This means that the rate of decay is not constant, but rather it increases as the quantity decreases.

Q: How do I determine whether a function represents exponential growth or decay?

A: To determine whether a function represents exponential growth or decay, you need to examine its general form. If the function has the form $y = ab^x$, where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time variable, and $b > 1$, the function represents exponential growth. If $0 < b < 1$, the function represents exponential decay.

Q: What is the base of an exponential function?

A: The base of an exponential function is the number that is raised to the power of the time variable. In the function $y = ab^x$, the base is $b$.

Q: How do I write the base in terms of the rate of growth or decay?

A: To write the base in terms of the rate of growth or decay, you need to identify the rate of growth or decay. If the function represents exponential growth, the base is the growth factor. If the function represents exponential decay, the base is the decay factor.

Q: What is the time variable in an exponential function?

A: The time variable in an exponential function is the variable that represents the time period over which the growth or decay occurs. In the function $y = ab^x$, the time variable is $x$.

Q: How do I interpret the rate of growth or decay?

A: To interpret the rate of growth or decay, you need to understand the meaning of the base. If the function represents exponential growth, the base represents the growth factor. If the function represents exponential decay, the base represents the decay factor.

Q: Can I use logarithms to solve exponential growth and decay problems?

A: Yes, you can use logarithms to solve exponential growth and decay problems. Logarithms can help you simplify the function and make it easier to solve.

Q: What are some real-world applications of exponential growth and decay?

A: Exponential growth and decay have many real-world applications, including:

• Population growth and decline
• Radioactive decay
• Compound interest
• Epidemiology

Q: How do I solve exponential growth and decay problems?

A: To solve exponential growth and decay problems, you need to identify the initial value, the growth or decay factor, and the time variable. You can then use the general form of an exponential function to solve the problem.

Q: What are some common mistakes to avoid when working with exponential growth and decay?

A: Some common mistakes to avoid when working with exponential growth and decay include:

• Confusing the base with the growth or decay factor
• Failing to identify the time variable
• Using the wrong formula for exponential growth or decay
• Not checking the units of the variables

Q: Can I use technology to solve exponential growth and decay problems?

A: Yes, you can use technology to solve exponential growth and decay problems. Many calculators and computer software programs have built-in functions for exponential growth and decay.

Q: What are some tips for working with exponential growth and decay?

A: Some tips for working with exponential growth and decay include:

• Always check the units of the variables
• Use the correct formula for exponential growth or decay
• Identify the time variable and the growth or decay factor
• Use logarithms to simplify the function
• Check your work carefully to avoid mistakes.