MTH120 College Algebra Chapter 6.1

 6.1 Exponential Functions

Exponential functions are a type of mathematical function in which the independent variable is in the exponent, and they are often written in the form $�(�)=�\cdot {�}^{�}$, where:

• $�$ is the initial value (the value of the function when $�=0$).
• $�$ is the base, and it's a positive constant (usually greater than 1) that determines how the function grows or decays.
• $�$ is the independent variable.

Exponential functions exhibit rapid growth or decay, and they are commonly encountered in various real-world applications, such as population growth, compound interest, and radioactive decay.

Key properties and characteristics of exponential functions include:

1. Initial Value: The value of the function when $�=0$ is $�$, the initial value.

2. Growth or Decay: The base $�$ determines whether the function grows or decays. If $�>1$, the function grows exponentially. If $0<�<1$, it decays exponentially.

3. Exponential Growth: If the base $�$ is greater than 1, the function increases rapidly as $�$ gets larger. The larger $�$ is, the steeper the growth.

4. Exponential Decay: If the base $�$ is between 0 and 1, the function decreases rapidly as $�$ gets larger. The closer $�$ is to 0, the steeper the decay.

5. Asymptotes: Exponential functions have horizontal asymptotes. In the case of exponential growth, the asymptote is $�=0$, and in exponential decay, it's also $�=0$.

6. Continuous and Smooth: Exponential functions are continuous and smooth curves with no breaks or sharp corners.

7. Inverse Operation: The inverse operation of exponentiation is taking the logarithm. If you have $�=�\cdot {�}^{�}$, taking the natural logarithm (base $�$, denoted as $\ln$) of both sides gives you $\ln (�)=\ln (�)+�\ln (�)$, which can be written in the form of a linear equation.

Exponential functions are widely used in the natural and social sciences to model various processes, and they play a significant role in understanding exponential growth and decay phenomena. Understanding their properties and characteristics is essential for working with exponential functions and solving problems involving them.

Identifying exponential functions involves recognizing patterns and characteristics that are specific to this type of mathematical function. Here are some key features and clues that can help you identify exponential functions:

1. Form of the Equation: Exponential functions are typically written in the form $�(�)=�\cdot {�}^{�}$, where $�$ is the initial value, and $�$ is the base.

2. Base (b): The base of the function is a positive constant. If $�$ is greater than 1, the function represents exponential growth. If $0<�<1$, the function represents exponential decay.

3. Exponential Growth: In cases of exponential growth, as the independent variable $�$ increases, the function value $�(�)$ increases rapidly. The larger the base $�$ is, the steeper the growth.

4. Exponential Decay: For exponential decay, as $�$ increases, the function value $�(�)$ decreases rapidly. The closer the base $�$ is to 0, the steeper the decay.

5. Initial Value (a): The initial value $�$ is the value of the function when $�=0$. It represents the starting point or the value at the beginning of the process.

6. Continuous and Smooth: Exponential functions are continuous and smooth curves. They have no breaks, gaps, or sharp corners.

7. Asymptotes: Exponential functions have horizontal asymptotes. In the case of exponential growth, the asymptote is $�=0$, and in exponential decay, it's also $�=0$.

8. Rate of Change: The rate of change of an exponential function is proportional to the current value of the function. This is what leads to rapid growth or decay.

9. Logarithmic Relationship: The inverse operation of exponentiation is taking the logarithm. If you can rewrite the equation in the form of $\ln (�)=\ln (�)+�\ln (�)$, it may be an indicator of an exponential function.

10. Real-World Scenarios: Exponential functions are often used to model real-world phenomena such as population growth, compound interest, radioactive decay, and more. If you encounter a problem where a quantity is growing or decaying at a constant percentage rate, it's likely an exponential function.

When analyzing equations or data, look for these characteristics to determine if you are dealing with an exponential function. Additionally, understanding the behavior and properties of exponential functions will help you identify them more effectively.

An exponential function is a mathematical function that can be defined as:

$�(�)=�\cdot {�}^{�}$

Where:

• $�(�)$ is the value of the function at a given $�$.
• $�$ is the initial value or the value of the function when $�=0$.
• $�$ is the base of the exponent, and it is a positive constant (usually greater than 1).

Exponential functions exhibit rapid growth or decay. The value of the function $�(�)$ increases or decreases exponentially as $�$ changes. The base $�$ determines the rate of growth or decay:

• If $�>1$, the function represents exponential growth. As $�$ increases, the value of the function grows rapidly.
• If $0<�<1$, the function represents exponential decay. As $�$ increases, the value of the function decreases rapidly.

Here are some important characteristics of exponential functions:

1. Initial Value (a): The value of $�$ is the starting point of the function. When $�=0$, $�(�)$ equals $�$.

2. Exponential Growth: If $�>1$, the function represents exponential growth. The larger the value of $�$, the steeper the growth.

3. Exponential Decay: If $0<�<1$, the function represents exponential decay. The closer $�$ is to 0, the steeper the decay.

4. Continuous and Smooth: Exponential functions are continuous and smooth curves. They have no breaks or sharp corners.

5. Asymptotes: Exponential functions have horizontal asymptotes. In the case of exponential growth, the asymptote is $�=0$, and in exponential decay, it's also $�=0$.

6. Inverse Operation: The inverse operation of exponentiation is taking the logarithm. If you have $�=�\cdot {�}^{�}$, taking the natural logarithm (base $�$, denoted as $\ln$) of both sides gives you $\ln (�)=\ln (�)+�\ln (�)$, which can be written in the form of a linear equation.

Exponential functions are commonly used to model a wide range of real-world phenomena, such as population growth, compound interest, radioactive decay, and more. Understanding these functions and their properties is essential for analyzing exponential growth and decay processes and solving problems related to them.

An exponential function is a type of mathematical function that is defined by the equation:

$�(�)=�\cdot {�}^{�}$

Where:

• $�(�)$ represents the value of the function at the point $�$.
• $�$ is the initial value, which is the value of the function when $�=0$.
• $�$ is the base of the exponent, a positive constant greater than 1 that determines the rate of growth (if $�>1$) or decay (if $0<�<1$).

Exponential functions exhibit rapid and continuous growth or decay. The value of the function increases or decreases exponentially as $�$ changes. The base $�$ controls how quickly this happens.

Key characteristics of exponential functions include:

1. Initial Value (a): This is the value of the function when $�=0$. It represents the starting point or the value at the beginning of the process.

2. Exponential Growth: If $�>1$, the function represents exponential growth. As $�$ increases, the value of the function grows rapidly. The larger the value of $�$, the steeper the growth.

3. Exponential Decay: If $0<�<1$, the function represents exponential decay. As $�$ increases, the value of the function decreases rapidly. The closer $�$ is to 0, the steeper the decay.

4. Continuous and Smooth: Exponential functions are continuous and smooth curves. They have no breaks, gaps, or sharp corners.

5. Asymptotes: Exponential functions have horizontal asymptotes. In the case of exponential growth, the asymptote is $�=0$, and in exponential decay, it's also $�=0$.

Exponential functions are widely used to model various real-world phenomena, such as population growth, compound interest, radioactive decay, and more. They play a significant role in understanding exponential growth and decay processes and are essential for solving problems related to these phenomena.

Evaluating exponential functions involves finding the value of the function for a given input value of $�$. The general form of an exponential function is $�(�)=�\cdot {�}^{�}$, where $�$ is the initial value, and $�$ is the base.

To evaluate an exponential function for a specific value of $�$, follow these steps:

1. Identify the values: Identify the values of $�$, $�$, and the specific $�$ for which you want to evaluate the function.

2. Substitute the values: Plug these values into the exponential function $�(�)=�\cdot {�}^{�}$.

3. Calculate: Use the order of operations (PEMDAS/BODMAS) to calculate the result.

Here's an example:

Example: Evaluate the exponential function $�(�)=2\cdot 3^{�}$ for $�=2$.

1. Identify the values:

   • $�=2$ (initial value)
   • $�=3$ (base)
   • $�=2$ (value for which we want to evaluate the function)

2. Substitute the values into the function:

   • $�(2)=2\cdot 3^2$

3. Calculate:

   • $�(2)=2\cdot 9$
   • $�(2)=18$

So, $�(2)=18$.

In this example, we've evaluated the exponential function $�(�)=2\cdot 3^{�}$ for $�=2$ and found that $�(2)=18$. This means that when $�=2$, the value of the function is 18. You can follow the same steps to evaluate exponential functions for different values of $�$.

The general form of an exponential function is:

$�(�)=�\cdot {�}^{�}$

Where:

• $�(�)$ is the function value at a given $�$.
• $�$ is the initial value or the value of the function when $�=0$.
• $�$ is the base of the exponent.
• $�$ is the exponent.

To evaluate an exponential function, you need to substitute a specific value for $�$ and perform the calculations. Here are the steps to evaluate an exponential function:

1. Identify the values of $�$, $�$, and $�$ in your specific exponential function.

2. Substitute the value of $�$ into the function.

3. Calculate ${�}^{�}$, which involves raising the base ($�$) to the power of $�$. You can use a calculator or mathematical software for this step.

4. Multiply the result from step 3 by $�$ to find the final function value $�(�)$.

Here's an example:

Let's say you have the exponential function $�(�)=2\cdot 3^4$, and you want to evaluate it at $�=4$.

1. Identify the values:

   • $�=2$
   • $�=3$
   • $�=4$

2. Substitute the value of $�$: $�(4)=2\cdot 3^4$

3. Calculate $3^4$: $3^4=3\cdot 3\cdot 3\cdot 3=81$

4. Multiply by $�$ to find $�(4)$: $�(4)=2\cdot 81=162$

So, $�(4)=162$.

This is how you evaluate an exponential function. You can use the same process for any exponential function with different values of $�$, $�$, and (x).

Exponential growth is a mathematical concept that describes a process in which a quantity or value increases at a constant relative rate over time. In other words, exponential growth occurs when a quantity multipliers by a fixed factor in each equal time period. The key characteristics of exponential growth are as follows:

1. Constant Percentage Rate: In exponential growth, the quantity increases by a fixed percentage or rate in each time interval. This means that the growth is proportional to the current value of the quantity. The larger the current quantity, the greater the increase in the next time period.

2. Compound Interest: Exponential growth is often associated with compound interest in finance. When you earn interest on an investment or have a debt with compound interest, your balance or debt increases by a certain percentage, resulting in exponential growth (if left unaltered).

3. Mathematical Representation: Exponential growth is typically represented by the following mathematical formula:

   $�(�)={�}_0\cdot {�}^{��}$

   Where:

   • $�(�)$ is the quantity at time $�$.
   • ${�}_0$ is the initial quantity (at $�=0$).
   • $�$ is the base of the natural logarithm (approximately 2.71828).
   • $�$ is the growth rate as a decimal (e.g., 0.05 for a 5% growth rate).
   • $�$ is time.

4. Rapid Growth: Exponential growth results in rapid increases over time, especially when compared to linear or other forms of growth. The growth becomes more pronounced as time progresses.

5. Unbounded: Exponential growth is unbounded, meaning there is no upper limit to how large the quantity can become. It can grow indefinitely if there are no limiting factors.

Examples of real-life situations exhibiting exponential growth include population growth (when resources are abundant), the spread of diseases, compound interest on investments, and the growth of microorganisms in favorable conditions.

It's important to note that exponential growth is an idealized concept and often doesn't continue indefinitely in real-world scenarios. Various factors, such as resource limitations, competition, or external constraints, can cause growth to slow down or level off, leading to a more realistic "logistic growth" pattern.

Evaluating a real-world exponential model involves applying the principles of exponential growth or decay to specific scenarios. Here are some examples to illustrate how to evaluate such models:

1. Population Growth:

   Example: A town's population starts with 10,000 people and is growing at a rate of 3% per year. Evaluate the population after 10 years.

   Model: $�(�)=10,000\cdot (1+0.03{)}^{10}$

   Evaluation: Calculate $�(10)=10,000\cdot (1.03{)}^{10}$ to find the population after 10 years.

2. Investment Growth:

   Example: You invest $5,000 in a savings account with an annual interest rate of 5%. Evaluate the value of your investment after 15 years.

   Model: $�(�)=5,000\cdot (1+0.05{)}^{15}$

   Evaluation: Calculate $�(15)=5,000\cdot (1.05{)}^{15}$ to find the value of your investment.

3. Radioactive Decay:

   Example: A sample of a radioactive isotope decays at a rate of 10% per hour. Evaluate the remaining amount of the isotope after 6 hours.

   Model: $�(�)={�}_0\cdot (0.9{)}^6$

   Evaluation: Calculate $�(6)={�}_0\cdot (0.9{)}^6$ to find the remaining amount of the isotope.

4. Spread of a Virus:

   Example: A virus is spreading exponentially. Initially, there are 100 infected individuals, and the number of infected individuals doubles every 2 days. Evaluate the number of infected individuals after 10 days.

   Model: $�(�)=100\cdot 2^{(�\mathrm{/}2)}$

   Evaluation: Calculate $�(10)=100\cdot 2^{(10\mathrm{/}2)}$ to find the number of infected individuals.

5. Exponential Depreciation:

   Example: A car's value depreciates at a rate of 15% per year. Evaluate the value of the car after 5 years.

   Model: $�(�)={�}_0\cdot (0.85{)}^5$

   Evaluation: Calculate $�(5)={�}_0\cdot (0.85{)}^5$ to find the car's value.

6. Bacterial Growth:

   Example: A colony of bacteria starts with 500 organisms and doubles in size every 30 minutes. Evaluate the number of bacteria after 2 hours.

   Model: $�(�)=500\cdot 2^{(�\mathrm{/}30)}$

   Evaluation: Calculate $�(120)=500\cdot 2^{(120\mathrm{/}30)}$ to find the number of bacteria.

In each of these examples, you apply the appropriate exponential growth or decay model, plug in the known values, and calculate the result. These models can be used for various real-world scenarios, such as finance, biology, and population studies, to make predictions and decisions based on the principles of exponential growth or decay.

To write an exponential model given two data points, you'll need to determine the values of the initial quantity (${�}_0$) and the growth/decay rate ($�$). The general form of an exponential model is:

$�(�)={�}_0\cdot {�}^{��}$

Where:

• $�(�)$ is the quantity at time $�$.
• ${�}_0$ is the initial quantity at $�=0$.
• $�$ is the base of the natural logarithm (approximately 2.71828).
• $�$ is the growth rate (positive for growth, negative for decay).
• $�$ is time.

Given two data points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$, you can set up a system of equations to solve for ${�}_0$ and $�$. Here's the system of equations:

1. ${�}_1={�}_0\cdot {�}^{�{�}_1}$
2. ${�}_2={�}_0\cdot {�}^{�{�}_2}$

Solve this system to find ${�}_0$ and $�$. Once you have these values, you can write the exponential model.

Here's a step-by-step process:

1. Calculate $�$ using the second equation: $�=\frac{\ln ({�}_2)-\ln ({�}_0)}{{�}_2-{�}_1}$

2. Substitute $�$ into the first equation to solve for ${�}_0$: ${�}_1={�}_0\cdot {�}^{�{�}_1}$ ${�}_0=\frac{{�}_1}{{�}^{�{�}_1}}$

Now that you have ${�}_0$ and $�$, you can write the exponential model using the general form:

$�(�)=\frac{{�}_1}{{�}^{�{�}_1}}\cdot {�}^{��}$

This model describes the exponential growth or decay of the quantity $�$ over time $�$ based on the given data points $({�}_1,{�}_1)$ and $({�}_2,{�}_2$).

To write the equation of an exponential function given its graph, you will need to determine the values of the initial quantity or y-intercept and the base of the exponent. The general form of an exponential function is:

$�(�)=�\cdot {�}^{�}$

Where:

• $�(�)$ is the function value at a given $�$.
• $�$ is the initial quantity or the value of the function when (x = 0.
• $�$ is the base of the exponent.
• $�$ is the independent variable.

Here's a step-by-step process to write the equation based on the graph:

1. Identify the y-intercept: This is the point where the graph intersects the y-axis, which corresponds to $�=0$. The y-coordinate of this point is the value of $�$, the initial quantity.

2. Determine the base ($�$): Look for any other point on the graph (different from the y-intercept) where both the x-coordinate ($�$) and the function value ($�(�)$) are known. Use these values to solve for $�$.

3. Write the equation: Once you have determined the values of $�$ and $�$, you can write the equation of the exponential function.

For example, let's say you have the following information from the graph:

• The y-intercept is at $(0,5)$.
• The graph passes through the point $(2,20)$.

Based on this information, you can write the equation of the exponential function as follows:

1. $�=5$ (from the y-intercept).

2. Use the second point $(2,20)$ to find $�$: $20=5\cdot {�}^2$

   Solve for $�$: ${�}^2=\frac{20}{5}=4$ $�=2$

3. Write the equation: $�(�)=5\cdot 2^{�}$

Now you have the equation of the exponential function based on the given graph. The graph's y-intercept provided the initial quantity ($�$), and another point on the graph helped you determine the base ($�$) of the exponential function.

To find the equation of an exponential function given two points on its curve using a graphing calculator, follow these steps:

1. Input the Data Points: Enter the coordinates of the two points into your graphing calculator. Most graphing calculators have a feature that allows you to input data points. These points will typically be in the form $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$.

2. Plot the Points: After entering the data points, plot them on the graphing calculator by selecting the appropriate function or graphing mode.

3. Determine the Exponential Equation: To find the exponential equation that fits the data, you can use the calculator's regression analysis feature. The exact steps may vary depending on the make and model of your calculator, but the general idea is as follows:

   • Access the calculator's statistical or regression functions (often found under STAT or similar menu options).
   • Choose the option for exponential regression (labeled EXP or similar).
   • Specify which data points to use (in this case, your two data points).
   • The calculator will perform the regression analysis and provide you with the equation of the exponential function that best fits the data.

4. Write Down the Equation: Once the calculator has determined the equation, it will typically be in the form:

   $�=�\cdot {�}^{�}$

   • $�$ will be the coefficient for the initial value or y-intercept.
   • $�$ will be the base of the exponent, representing the growth or decay factor.

   Write down the equation that the calculator provides.

5. Check the Equation: Verify that the equation fits the data points you initially input by plugging in the x-values and ensuring that the calculated y-values match the given y-values.

6. Use the Equation: With the equation in hand, you can use it to make predictions or analyze the behavior of the exponential function in the given context.

Please note that the specific steps and menu options on your graphing calculator may vary, so refer to your calculator's user manual for precise instructions on how to perform exponential regression and obtain the equation.

Applying the compound-interest formula is important when you want to calculate the future value of an investment or loan that compounds over time. Compound interest is a method of accruing interest not just on the initial principal (the initial amount of money), but also on the accumulated interest from previous periods. The formula for compound interest is:

$�=�{(1+\frac{�}{�})}^{��}$

Where:

• $�$ is the future value of the investment/loan, including interest.
• $�$ is the principal amount (initial investment or loan amount).
• $�$ is the annual interest rate (as a decimal).
• $�$ is the number of times that interest is compounded per year.
• $�$ is the number of years the money is invested or borrowed for.

Here are the steps to apply the compound-interest formula:

1. Identify the Values:

   • Determine the principal amount ($�$) or initial loan amount.
   • Find the annual interest rate ($�$) as a decimal.
   • Decide how often the interest is compounded per year and note $�$.
   • Specify the number of years ($�$) for which you want to calculate the future value.

2. Plug Values into the Formula:

   • Insert the values you identified into the compound-interest formula.
   • Make sure the interest rate ($�$) is expressed as a decimal, so divide the annual rate by 100 if necessary.

3. Calculate:

   • Use a calculator or spreadsheet to calculate $�$ (the future value).
   • The result represents the total amount you will have (if investing) or owe (if borrowing) after the specified time period.

4. Interpret the Result:

   • The calculated value $�$ is the future worth of your investment or the total amount owed on a loan after the specified time.

Here's an example:

Let's say you invest $5,000 in a savings account with an annual interest rate of 5%, compounded quarterly for 3 years. Calculate the future value of your investment.

• $�=5000$ (principal)
• $�=0.05$ (5% annual interest rate as a decimal)
• $�=4$ (quarterly compounding)
• $�=3$ (3 years)

Now, apply the formula:

$�=5000{(1+\frac{0.05}{4})}^{4\cdot 3}$

Calculate $�$, and you'll find the future value of your investment.

The compound interest formula is a mathematical formula used to calculate the future value of an investment or loan when interest is compounded periodically. It takes into account the initial principal amount, the annual interest rate, the number of times interest is compounded per year, and the number of years the money is invested or borrowed for.

The formula for compound interest is:

$�=�{(1+\frac{�}{�})}^{��}$

Where:

• $�$ is the future value of the investment or loan, including interest.
• $�$ is the principal amount (the initial amount of money).
• $�$ is the annual interest rate (as a decimal).
• $�$ is the number of times that interest is compounded per year.
• $�$ is the number of years the money is invested or borrowed for.

Here's a breakdown of what each variable represents:

• $�$ is the amount of money you will have or owe in the future, including both the principal and accumulated interest.
• $�$ is the initial principal amount, the amount you start with.
• $�$ is the annual interest rate, expressed as a decimal (e.g., 5% as 0.05).
• $�$ is the number of times that interest is compounded per year. If it's compounded annually, $�=1$; if semi-annually, $�=2$; quarterly, $�=4$; and so on.
• $�$ is the number of years the money is invested or borrowed for.

The formula calculates the future value $�$ based on the initial principal, the interest rate, and how often interest is compounded. Compound interest allows for the interest earned or paid to be added or charged more frequently, and this results in a higher final amount compared to simple interest.

This formula is commonly used in financial calculations, such as savings accounts, loans, mortgages, and investments, to determine the future value of money over time. It's essential to understand the concept of compound interest when managing your finances or making financial decisions.

You can use the compound interest formula to solve for the principal amount ($�$) when you have all the other variables in the formula. The compound interest formula is:

$�=�{(1+\frac{�}{�})}^{��}$

To solve for the principal ($�$), you need the following information:

1. The future value ($�$) of the investment or loan.
2. The annual interest rate ($�$) as a decimal.
3. The number of times interest is compounded per year ($�$).
4. The number of years ($�$) the money is invested or borrowed for.

Here's how to solve for $�$:

1. Rearrange the formula to isolate $�$:

   $�=\frac{�}{{(1+\frac{�}{�})}^{��}}$

2. Plug in the values for $�$, $�$, $�$, and $�$ into the formula.

3. Calculate the result to find the principal ($�$).

Let's go through an example:

Suppose you want to find the principal amount needed to have $10,000 in a savings account after 5 years with an annual interest rate of 6%, compounded quarterly ($�=4$).

1. Values:

   • $�=10,000$ (the future value)
   • $�=0.06$ (6% as a decimal)
   • $�=4$ (quarterly compounding)
   • $�=5$ (5 years)

2. Apply the formula:

   $�=\frac{10,000}{{(1+\frac{0.06}{4})}^{4\cdot 5}}$

3. Calculate $�$:

   $�=\frac{10,000}{{(1+0.015)}^{20}}$ $�=\frac{10,000}{(1.015{)}^{20}}$

You can calculate the value of $�$ to find the initial principal required to reach $10,000 after 5 years with the given interest rate and compounding frequency.

When evaluating functions with base $�$, you are dealing with exponential functions where the base is Euler's number, denoted as $�$, approximately equal to 2.71828. The general form of such a function is:

$�(�)=�\cdot {�}^{��}$

Here's how you can evaluate a function with base $�$:

1. Identify the Parameters:

   • $�$ is the initial value or the value when $�=0$.
   • $�$ is the constant that determines the rate of growth or decay.
   • $�$ is the variable.

2. Substitute the Value of $�$: Substitute the specific value of $�$ into the function.

   $�(�)=�\cdot {�}^{��}$

3. Calculate ${�}^{��}$: Evaluate the exponential term ${�}^{��}$ by raising $�$ to the power of $��$. Use a calculator or mathematical software for this calculation.

4. Multiply by $�$: Multiply the result from step 3 by the coefficient $�$ to find the final function value $�(�)$.

Here's an example:

Let's say you have the function $�(�)=2\cdot {�}^{0.5�}$, and you want to evaluate it at $�=3$.

1. Identify the values:

   • $�=2$
   • $�=0.5$
   • $�=3$

2. Substitute the value of $�$: $�(3)=2\cdot {�}^{0.5\cdot 3}$

3. Calculate ${�}^{0.5\cdot 3}$: ${�}^{0.5\cdot 3}={�}^{1.5}$

4. Multiply by $�$ to find $�(3)$: $�(3)=2\cdot {�}^{1.5}$

You can use a calculator to find the numerical value of ${�}^{1.5}$ and then multiply by 2 to obtain the final result for $�(3)$.

Euler's number ($�$) is a fundamental constant in mathematics, and exponential functions based on $�$ are common in various scientific and financial applications. The evaluation of such functions helps in making predictions and understanding growth or decay in different contexts.

You can use a calculator to find powers of the mathematical constant $�$ (Euler's number) or to evaluate exponential expressions involving $�$. Most calculators, including scientific and graphing calculators, have a built-in function for calculating ${�}^{�}$ for any value of $�$. Here's how to do it:

1. Turn on your calculator: Ensure that your calculator is turned on and ready for use.

2. Access the exponential function: On most calculators, you'll find the exponential function as "e^x" or "exp(x)." It's typically located near other trigonometric or logarithmic functions.

3. Input the exponent ($�$): Enter the value of the exponent you want to evaluate. For example, if you want to find ${�}^2$, enter "2."

4. Calculate: Press the "equals" or "calculate" button on your calculator. This will compute the value of ${�}^{�}$ for the given exponent.

For example, to calculate ${�}^2$, you would:

• On most calculators, press "e^x" or "exp(x."
• Input "2" as the exponent.
• Press the "equals" button, and the calculator will display the result, which is approximately 7.389056.

If you need to evaluate other values, simply change the exponent ($�$) accordingly.

Keep in mind that many calculators also offer additional functions for computing natural logarithms (ln), which can be helpful when working with expressions involving $�$. For instance, if you need to calculate the natural logarithm of ${�}^{�}$, you can do so by taking the ln of the result:

• Calculate ${�}^{�}$ as described above.
• Then, calculate the natural logarithm (ln) of the result to find $\ln ({�}^{�})$, which should be equal to $�$.

Using your calculator, you can quickly and accurately perform calculations involving the mathematical constant $�$.

Continuous growth is a mathematical concept that describes a process where a quantity increases or decreases continuously over time. It is often associated with exponential growth or decay. Continuous growth occurs when a quantity changes at every infinitesimally small moment in time. This is in contrast to discrete growth, which occurs at specific, discrete time intervals.

To investigate continuous growth, you can follow these steps:

1. Understand the Exponential Growth/Decay Model: Continuous growth is typically described by an exponential function, which can be written in the form:

   $�(�)={�}_0\cdot {�}^{��}$

   • $�(�)$ is the quantity at time $�$.
   • ${�}_0$ is the initial quantity at $�=0$.
   • $�$ is Euler's number (approximately 2.71828).
   • $�$ is the continuous growth or decay rate.
   • $�$ is time.

2. Identify the Parameters:

   • Determine the initial quantity (${�}_0$).
   • Find the continuous growth or decay rate ($�$).

3. Evaluate and Interpret:

   • Calculate the value of the function for various values of $�$ to understand how the quantity changes continuously over time.
   • Interpret the results to gain insights into the growth or decay process.

4. Examine Limiting Behavior:

   • Investigate the limiting behavior of the continuous growth process. Determine whether the growth approaches infinity, remains bounded, or tends to zero over time.

5. Apply in Real-World Scenarios:

   • Apply the concept of continuous growth to real-world scenarios, such as population growth, financial investments, radioactive decay, and more.

6. Sensitivity Analysis:

   • Analyze how variations in the continuous growth rate ($�$) impact the behavior of the system. A small change in $�$ can lead to significant differences in the outcome.

7. Use Mathematical Tools:

   • Utilize mathematical tools, such as calculus, to investigate continuous growth in greater detail, including finding the derivative of the function to determine the rate of change.

Continuous growth is a fundamental concept in mathematics and has many applications in various fields. It allows for a more accurate modeling of processes that change continuously over time, rather than in discrete steps. Understanding continuous growth is essential in making predictions, analyzing data, and solving real-world problems.

For the following exercises, find the formula for an exponential function that passes through the two points given. 1. (0,4) and (2,750) 2. (0,1000) and (1,22) 3. (−1,22) and (3,44) 4. (−1,7) and (2,1) 5. (6,1) and (3,8)

To find the formula for an exponential function that passes through two given points, you can use the general exponential function form:

$�(�)=�\cdot {�}^{�}$

where $�$ is the initial value at $�=0$, and $�$ is the base of the exponential growth or decay. Given two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$, you can set up a system of equations and solve for $�$ and $�$ in each case. Here are the solutions for the given points:

1. (0, 4) and (2, 750):

   • Using $(0,4)$: $4=�\cdot {�}^0$ $4=�$
   • Using $(2,750)$: $750=4\cdot {�}^2$ ${�}^2=\frac{750}{4}=187.5$ $�=\sqrt{187.5}\approx 13.69$

   So, the formula for the exponential function is: $�(�)=4\cdot 13.69^{�}$

2. (0, 1000) and (1, 22):

   • Using $(0,1000)$: $1000=�\cdot {�}^0$ $1000=�$
   • Using $(1,22)$: $22=1000\cdot {�}^1$ $�=\frac{22}{1000}=0.022$

   The exponential function is: $�(�)=1000\cdot 0.022^{�}$

3. (−1, 22) and (3, 44):

   • Using $(-1,22)$: $22=�\cdot {�}^{-1}$ $22=\frac{�}{�}$ $�=22�$
   • Using $(3,44)$: $44=22�\cdot {�}^3$ $44=22{�}^4$ ${�}^4=\frac{44}{22}=2$ $�=\sqrt[4]{2}\approx 1.4142$

   The exponential function is: $�(�)=22\cdot (1.4142{)}^{�}$

4. (−1, 7) and (2, 1):

   • Using $(-1,7)$: $7=�\cdot {�}^{-1}$ $7=\frac{�}{�}$ $�=7�$
   • Using $(2,1)$: $1=7�\cdot {�}^2$ $1=7{�}^3$ ${�}^3=\frac{1}{7}$ $�=\sqrt[3]{\frac{1}{7}}$

   The exponential function is: $�(�)=7\cdot {\left(\sqrt[3]{\frac{1}{7}}\right)}^{�}$

5. (6, 1) and (3, 8):

   • Using $(6,1)$: $1=�\cdot {�}^6$
   • Using $(3,8)$: $8=�\cdot {�}^3$

   Divide the second equation by the first: $\frac{8}{1}=\frac{�\cdot {�}^3}{�\cdot {�}^6}$ $8={�}^{-3}$ $�=\frac{1}{8}=0.125$

   The exponential function is: $�(�)=�\cdot (0.125{)}^{�}$

You now have the formulas for exponential functions that pass through the given points.

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. 7. x 1 2 3 4 f(x) 60 20 11 -21 8. x 1 2 3 4 h(x) 40 47 35.3 24.01 9. x 1 2 3 4 m(x) 90 51 42.9 24.71 10. x 1 2 3 4 f(x) 50 70 60 70 11. x

1 2 3 4 g(x) -2.25 3 7.55 22.5

Let's analyze each table and determine whether the given data appears to represent a linear, exponential, or neither function. If it seems exponential, we'll find the corresponding function.

7. Table for $�(�)$:

   $$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}\\ {\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span>}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">60</span>}\mathrm{<span\; style="background-color:\; white;">20</span>}\mathrm{<span\; style="background-color:\; white;">11</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">21</span>}}\end{array}$$

   This table appears to represent an exponential decay. Let's find the function. $�(�)=60\cdot {\left(\frac{1}{3}\right)}^{�}$

8. Table for $ℎ(�)$:

   $$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}\\ {\displaystyle \mathrm{<span\; style="background-color:\; white;">ℎ</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span>}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">40</span>}\mathrm{<span\; style="background-color:\; white;">47</span>}\mathrm{<span\; style="background-color:\; white;">35.3</span>}\mathrm{<span\; style="background-color:\; white;">24.01</span>}}\end{array}$$

   This table seems neither linear nor exponential, as there is no consistent pattern in the ratios of consecutive terms.

9. Table for $�(�)$:

   $$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}\\ {\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span>}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">90</span>}\mathrm{<span\; style="background-color:\; white;">51</span>}\mathrm{<span\; style="background-color:\; white;">42.9</span>}\mathrm{<span\; style="background-color:\; white;">24.71</span>}}\end{array}$$

   This table appears to represent an exponential decay. $�(�)=90\cdot {\left(\frac{3}{5}\right)}^{�}$

10. Table for $�(�)$:

$$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}\\ {\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span>}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">50</span>}\mathrm{<span\; style="background-color:\; white;">70</span>}\mathrm{<span\; style="background-color:\; white;">60</span>}\mathrm{<span\; style="background-color:\; white;">70</span>}}\end{array}$$

This table does not represent a clear linear or exponential pattern. It's neither.

11. Table for $�(�)$:

$$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}}& {\displaystyle <span\; style="background-color:\; white;">:</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}\\ {\displaystyle \mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span>}& {\displaystyle <span\; style="background-color:\; white;">:</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">2.25</span>}\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">7.55</span>}\mathrm{<span\; style="background-color:\; white;">22.5</span>}}\end{array}$$

This table appears to represent an exponential growth. $�(�)=-2.25\cdot {�}^{1.5�}$

In summary:

• Tables 7 and 9 represent exponential decay.
• Table 8 does not appear to represent a clear pattern.
• Tables 10 and 11 do not represent a clear linear or exponential pattern.