Write An Exponential Function In The Form Y = A B X Y = Ab^x Y = A B X That Goes Through The Points ( 0 , 9 (0, 9 ( 0 , 9 ] And ( 5 , 2187 (5, 2187 ( 5 , 2187 ].

Introduction

Exponential functions are a fundamental concept in mathematics, used to model real-world phenomena that exhibit rapid growth or decay. In this article, we will explore how to write an exponential function in the form $y = ab^x$ that passes through the points $(0, 9)$ and $(5, 2187)$.

What are Exponential Functions?

Exponential functions are a type of mathematical function that can be written in the form $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is a positive real number, and the exponent $x$ is a real number. Exponential functions are used to model situations where a quantity grows or decays at a constant rate.

The General Form of an Exponential Function

The general form of an exponential function is $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is a positive real number, and the exponent $x$ is a real number. The value of $a$ determines the initial value of the function, while the value of $b$ determines the rate of growth or decay.

Finding the Exponential Function that Passes Through the Given Points

To find the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$, we need to use the given points to set up a system of equations. We can start by substituting the values of $x$ and $y$ from the given points into the general form of the exponential function.

Step 1: Substitute the Values of $x$ and $y$ from the Given Points

Let's substitute the values of $x$ and $y$ from the given points into the general form of the exponential function.

• For the point $(0, 9)$, we have $y = 9$ and $x = 0$. Substituting these values into the general form of the exponential function, we get: $9 = ab^0$
• For the point $(5, 2187)$, we have $y = 2187$ and $x = 5$. Substituting these values into the general form of the exponential function, we get: $2187 = ab^5$

Step 2: Simplify the Equations

Now that we have substituted the values of $x$ and $y$ from the given points into the general form of the exponential function, we can simplify the equations.

• For the point $(0, 9)$, we have $9 = ab^0$. Since $b^0 = 1$, we can simplify this equation to: $9 = a$
• For the point $(5, 2187)$, we have $2187 = ab^5$. We can substitute the value of $a$ from the previous equation into this equation to get: $2187 = 9b^5$

Step 3: Solve for $b$

Now that we have simplified the equations, we can solve for $b$.

• From the equation $2187 = 9b^5$, we can divide both sides by 9 to get: $243 = b^5$
• Taking the fifth root of both sides, we get: $b = 3$

Step 4: Write the Exponential Function

Now that we have found the value of $b$, we can write the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$.

• Since $a = 9$ and $b = 3$, we can substitute these values into the general form of the exponential function to get: $y = 9(3^x)$

Conclusion

In this article, we have shown how to write an exponential function in the form $y = ab^x$ that passes through the points $(0, 9)$ and $(5, 2187)$. We have used the given points to set up a system of equations, simplified the equations, and solved for $b$. Finally, we have written the exponential function that passes through the given points.

Exponential Functions in Real-World Applications

Exponential functions have many real-world applications, including:

• Population growth: Exponential functions can be used to model population growth, where the population grows at a constant rate.
• Compound interest: Exponential functions can be used to model compound interest, where the interest rate is applied to the principal amount at regular intervals.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at a constant rate.

Common Mistakes to Avoid

When working with exponential functions, there are several common mistakes to avoid:

• Incorrectly substituting values: Make sure to substitute the correct values of $x$ and $y$ from the given points into the general form of the exponential function.
• Simplifying equations incorrectly: Make sure to simplify the equations correctly, and avoid making mistakes when solving for $b$.
• Writing the exponential function incorrectly: Make sure to write the exponential function correctly, using the correct values of $a$ and $b$.

Final Thoughts

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Q&A: Exponential Functions

Q: What is an exponential function?

A: An exponential function is a type of mathematical function that can be written in the form $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is a positive real number, and the exponent $x$ is a real number.

Q: What is the general form of an exponential function?

A: The general form of an exponential function is $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is a positive real number, and the exponent $x$ is a real number.

Q: How do I find the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$?

A: To find the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$, you need to use the given points to set up a system of equations. You can start by substituting the values of $x$ and $y$ from the given points into the general form of the exponential function.

Q: What are the steps to find the exponential function that passes through the given points?

A: The steps to find the exponential function that passes through the given points are:

1. Substitute the values of $x$ and $y$ from the given points into the general form of the exponential function.
2. Simplify the equations.
3. Solve for $b$.
4. Write the exponential function.

Q: What is the value of $b$ in the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$?

A: The value of $b$ in the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$ is $3$.

Q: What is the exponential function that passes through the points $(0, 9)$ and $(5, 2187)$?

A: The exponential function that passes through the points $(0, 9)$ and $(5, 2187)$ is $y = 9(3^x)$.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including:

• Population growth
• Compound interest
• Radioactive decay

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

• Incorrectly substituting values
• Simplifying equations incorrectly
• Writing the exponential function incorrectly

Q: How can I use exponential functions to model real-world phenomena?

A: You can use exponential functions to model real-world phenomena by:

• Using the general form of the exponential function to model the phenomenon
• Substituting the values of $x$ and $y$ from the given data into the general form of the exponential function
• Simplifying the equations
• Solving for $b$
• Writing the exponential function

Q: What are some tips for working with exponential functions?

A: Some tips for working with exponential functions include:

• Make sure to substitute the correct values of $x$ and $y$ from the given points into the general form of the exponential function.
• Simplify the equations correctly, and avoid making mistakes when solving for $b$.
• Write the exponential function correctly, using the correct values of $a$ and $b$.

Conclusion

Exponential functions are a powerful tool for modeling real-world phenomena that exhibit rapid growth or decay. By following the steps outlined in this article, you can write an exponential function in the form $y = ab^x$ that passes through the points $(0, 9)$ and $(5, 2187)$. Remember to avoid common mistakes, and to use exponential functions to model real-world applications.