Write An Exponential Function In The Form Y = A B X Y = Ab^x Y = A B X That Goes Through The Points ( 0 , 8 (0,8 ( 0 , 8 ] And ( 2 , 288 (2,288 ( 2 , 288 ].

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. In this article, we will focus on writing an exponential function in the form $y = ab^x$ that passes through the points $(0,8)$ and $(2,288)$.

What are Exponential Functions?

Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the independent variable. The constant $a$ is called the initial value or the y-intercept, and it represents the value of the function when $x = 0$. The constant $b$ is called the base, and it determines the rate at which the function grows or decays.

Properties of Exponential Functions

Exponential functions have several important properties that make them useful in modeling real-world phenomena. Some of the key properties of exponential functions include:

• Exponential growth: Exponential functions can grow or decay at an exponential rate, which means that the value of the function increases or decreases rapidly as $x$ increases.
• Initial value: The initial value of an exponential function is the value of the function when $x = 0$. This value is represented by the constant $a$.
• Base: The base of an exponential function is the constant $b$, which determines the rate at which the function grows or decays.
• Domain and range: The domain of an exponential function is all real numbers, and the range is all positive real numbers.

Writing an Exponential Function

To write an exponential function in the form $y = ab^x$ that passes through the points $(0,8)$ and $(2,288)$, we need to find the values of $a$ and $b$. We can use the given points to create a system of equations and solve for $a$ and $b$.

Step 1: Create a System of Equations

We can use the given points to create a system of equations:

• Equation 1: $8 = ab^0$
• Equation 2: $288 = ab^2$

Step 2: Simplify the Equations

We can simplify the equations by using the fact that $b^0 = 1$ and $b^2 = b \cdot b$.

• Equation 1: $8 = a \cdot 1$
• Equation 2: $288 = a \cdot b \cdot b$

Step 3: Solve for $a$ and $b$

We can solve for $a$ and $b$ by dividing Equation 2 by Equation 1:

• Equation 3: $\frac{288}{8} = \frac{a \cdot b \cdot b}{a \cdot 1}$
• Equation 4: $36 = b^2$

Step 4: Find the Value of $b$

We can find the value of $b$ by taking the square root of both sides of Equation 4:

• Equation 5: $b = \sqrt{36}$
• Equation 6: $b = 6$

Step 5: Find the Value of $a$

We can find the value of $a$ by substituting the value of $b$ into Equation 1:

• Equation 7: $8 = a \cdot 1$
• Equation 8: $a = 8$

Conclusion

In this article, we have written an exponential function in the form $y = ab^x$ that passes through the points $(0,8)$ and $(2,288)$. We have used the given points to create a system of equations and solved for $a$ and $b$. The final exponential function is $y = 8 \cdot 6^x$.

Exponential Function in the Form $y = ab^x$

$y = 8 \cdot 6^x$

Graph of the Exponential Function

The graph of the exponential function $y = 8 \cdot 6^x$ is a curve that passes through the points $(0,8)$ and $(2,288)$. The graph shows exponential growth, with the value of the function increasing rapidly as $x$ increases.

Applications of Exponential Functions

Exponential functions have numerous applications in various fields, including science, engineering, and economics. Some of the key applications of exponential functions include:

• Population growth: Exponential functions can be used to model population growth, where the population increases at an exponential rate.
• Compound interest: Exponential functions can be used to calculate compound interest, where the interest rate is applied at an exponential rate.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at an exponential rate.

Conclusion

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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 describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y = ab^x$, where $a$ and $b$ are constants, and $x$ is the independent variable.

Q: What are the properties of exponential functions?

A: Exponential functions have several important properties that make them useful in modeling real-world phenomena. Some of the key properties of exponential functions include:

• Exponential growth: Exponential functions can grow or decay at an exponential rate, which means that the value of the function increases or decreases rapidly as $x$ increases.
• Initial value: The initial value of an exponential function is the value of the function when $x = 0$. This value is represented by the constant $a$.
• Base: The base of an exponential function is the constant $b$, which determines the rate at which the function grows or decays.
• Domain and range: The domain of an exponential function is all real numbers, and the range is all positive real numbers.

Q: How do I write an exponential function in the form $y = ab^x$?

A: To write an exponential function in the form $y = ab^x$, you need to find the values of $a$ and $b$. You can use the given points to create a system of equations and solve for $a$ and $b$.

Q: What is the difference between exponential growth and decay?

A: Exponential growth occurs when the value of the function increases rapidly as $x$ increases, while exponential decay occurs when the value of the function decreases rapidly as $x$ increases.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program to plot the function. You can also use a table of values to create a graph by hand.

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

A: Exponential functions have numerous applications in various fields, including science, engineering, and economics. Some of the key applications of exponential functions include:

• Population growth: Exponential functions can be used to model population growth, where the population increases at an exponential rate.
• Compound interest: Exponential functions can be used to calculate compound interest, where the interest rate is applied at an exponential rate.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at an exponential rate.

Q: How do I solve exponential equations?

A: To solve exponential equations, you can use the following steps:

1. Isolate the exponential term: Move all terms except the exponential term to one side of the equation.
2. Take the logarithm of both sides: Take the logarithm of both sides of the equation to eliminate the exponential term.
3. Solve for the variable: Solve for the variable by isolating it on one side of the equation.

Q: What is the difference between exponential and linear functions?

A: Exponential functions grow or decay at an exponential rate, while linear functions grow or decay at a constant rate.

Q: How do I determine if a function is exponential or linear?

A: To determine if a function is exponential or linear, you can use the following steps:

1. Graph the function: Graph the function to see if it is a straight line or a curve.
2. Check the rate of change: Check the rate of change of the function to see if it is constant or changing rapidly.
3. Check the equation: Check the equation of the function to see if it is in the form $y = ab^x$ or $y = mx + b$.

Conclusion

In conclusion, exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have answered some common questions about exponential functions, including how to write an exponential function in the form $y = ab^x$, how to graph an exponential function, and how to solve exponential equations. We hope that this article has been helpful in understanding exponential functions and their applications.