An Exponential Function $f(x)=a \cdot B^x$ Passes Through The Points $(0,2)$ And \$(3,54)$[/tex\]. What Are The Values Of $a$ And $b$?$a=$ $\square$ And $b=$

Feb 28, 2025 by ADMIN 167 views

An Exponential Function: Finding the Values of a and b

In mathematics, an exponential function is a function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants. The value of $a$ represents the initial value of the function, while the value of $b$ represents the growth factor. In this article, we will explore how to find the values of $a$ and $b$ given two points through which the exponential function passes.

We are given two points through which the exponential function passes: $(0, 2)$ and $(3, 54)$ . Our goal is to find the values of $a$ and $b$ that satisfy the equation $f(x) = a \cdot b^x$ for these two points.

Let's start by using the first point $(0, 2)$ to find the value of $a$ . We know that when $x = 0$ , the value of the function is $2$ . Substituting these values into the equation, we get:

f(0) = a \cdot b^0 = 2

Since any number raised to the power of $0$ is equal to $1$ , we can simplify the equation to:

a \cdot 1 = 2

This implies that $a = 2$ .

Now that we have found the value of $a$ , let's use the second point $(3, 54)$ to find the value of $b$ . We know that when $x = 3$ , the value of the function is $54$ . Substituting these values into the equation, we get:

f(3) = a \cdot b^3 = 54

Since we have already found that $a = 2$ , we can substitute this value into the equation to get:

2 \cdot b^3 = 54

Dividing both sides of the equation by $2$ , we get:

b^3 = 27

Taking the cube root of both sides of the equation, we get:

b = 3

In this article, we have used two points through which an exponential function passes to find the values of $a$ and $b$ . We found that $a = 2$ and $b = 3$ . These values satisfy the equation $f(x) = a \cdot b^x$ for the given points.

Exponential functions are used to model a wide range of phenomena in mathematics, science, and engineering. They are used to describe population growth, chemical reactions, and electrical circuits, among other things. Understanding how to find the values of $a$ and $b$ in an exponential function is an important skill that can be applied to a variety of real-world problems.

Exponential functions have many real-world applications. For example, they are used to model population growth, where the population of a species grows exponentially over time. They are also used to model chemical reactions, where the concentration of a substance changes exponentially over time. In addition, exponential functions are used to model electrical circuits, where the voltage and current change exponentially over time.

In conclusion, exponential functions are an important concept in mathematics that have many real-world applications. Understanding how to find the values of $a$ and $b$ in an exponential function is an important skill that can be applied to a variety of problems. By following the steps outlined in this article, you can find the values of $a$ and $b$ given two points through which the exponential function passes.

The final answer is:

In our previous article, we explored how to find the values of $a$ and $b$ in an exponential function given two points through which the function passes. In this article, we will answer some frequently asked questions about exponential functions and provide additional examples and explanations.

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants. The value of $a$ represents the initial value of the function, while the value of $b$ represents the growth factor.

Q: How do I find the values of $a$ and $b$ in an exponential function?

A: To find the values of $a$ and $b$ in an exponential function, you can use two points through which the function passes. You can use the first point to find the value of $a$ and the second point to find the value of $b$ .

Q: What is the difference between an exponential function and a linear function?

A: An exponential function grows or decays at a rate that is proportional to the current value of the function, while a linear function grows or decays at a constant rate. For example, if you have a linear function that represents the cost of a product, the cost will increase by a fixed amount for each unit sold. However, if you have an exponential function that represents the cost of a product, the cost will increase by a percentage of the current cost for each unit sold.

Q: Can I use an exponential function to model a population that is decreasing?

A: Yes, you can use an exponential function to model a population that is decreasing. In this case, the value of $b$ will be less than 1, indicating that the population is decreasing.

Q: How do I graph an exponential function?

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

Q: Can I use an exponential function to model a chemical reaction?

A: Yes, you can use an exponential function to model a chemical reaction. In this case, the value of $b$ will represent the rate at which the reaction occurs.

Q: What is the significance of the value of $b$ in an exponential function?

A: The value of $b$ in an exponential function represents the growth factor, which determines how quickly the function grows or decays. A value of $b$ greater than 1 indicates that the function is growing, while a value of $b$ less than 1 indicates that the function is decaying.

Q: Can I use an exponential function to model a financial investment?

A: Yes, you can use an exponential function to model a financial investment. In this case, the value of $b$ will represent the rate of return on the investment.

Q: How do I determine the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, while the range is all positive real numbers.

In conclusion, exponential functions are a powerful tool for modeling a wide range of phenomena in mathematics, science, and engineering. By understanding how to find the values of $a$ and $b$ in an exponential function, you can apply this knowledge to a variety of real-world problems.

Here are a few additional examples of exponential functions:

$f(x) = 2 \cdot 3^x$
$f(x) = 5 \cdot 2^x$
$f(x) = 10 \cdot 0.5^x$