The Table Shows Two Points On The Graph Of An Exponential Function Of The Form Y = A B X Y = Ab^x Y = A B X , Where B \textgreater 1 B \ \textgreater \ 1 B \textgreater 1 .${ \begin{array}{|c|c|} \hline x & Y \ \hline 1 & 1 \ \hline 3 & 16 \ \hline \end{array} }$What Is

Introduction

In mathematics, an exponential function is a function that has the form $y = ab^x$, where $a$ and $b$ are constants and $b > 1$. These functions are used to model a wide range of phenomena, from population growth to chemical reactions. In this article, we will explore how to find the values of $a$ and $b$ given two points on the graph of an exponential function.

The Given Points

The table shows two points on the graph of an exponential function:

x	y
1	1
3	16

Using the Points to Find the Values of $a$ and $b$

To find the values of $a$ and $b$, we can use the fact that the function is of the form $y = ab^x$. We can start by using the first point $(1, 1)$ to find the value of $a$. Since $y = ab^x$, we can substitute $x = 1$ and $y = 1$ to get:

$$1 = a \cdot b^1$$

Simplifying this equation, we get:

$$a = \frac{1}{b}$$

Now, we can use the second point $(3, 16)$ to find the value of $b$. Substituting $x = 3$ and $y = 16$ into the equation $y = ab^x$, we get:

$$16 = a \cdot b^3$$

Substituting the expression for $a$ that we found earlier, we get:

$$16 = \frac{1}{b} \cdot b^3$$

Simplifying this equation, we get:

$$16 = b^2$$

Taking the square root of both sides, we get:

$$b = \pm 4$$

Since $b > 1$, we know that $b = 4$.

Finding the Value of $a$

Now that we have found the value of $b$, we can find the value of $a$. Substituting $b = 4$ into the expression $a = \frac{1}{b}$, we get:

$$a = \frac{1}{4}$$

The Final Answer

Therefore, the values of $a$ and $b$ are $a = \frac{1}{4}$ and $b = 4$.

Conclusion

In this article, we have shown how to find the values of $a$ and $b$ given two points on the graph of an exponential function. We used the fact that the function is of the form $y = ab^x$ and substituted the values of $x$ and $y$ into the equation to find the values of $a$ and $b$. We also used the fact that $b > 1$ to determine that $b = 4$. Finally, we found the value of $a$ by substituting $b = 4$ into the expression $a = \frac{1}{b}$.

Exponential Functions in Real-World Applications

Exponential functions have many real-world applications, including:

• Population growth: Exponential functions can be used to model the growth of populations over time.
• Chemical reactions: Exponential functions can be used to model the rate of chemical reactions.
• Finance: Exponential functions can be used to model the growth of investments over time.
• Biology: Exponential functions can be used to model the growth of bacteria and other microorganisms.

Examples of Exponential Functions

Some examples of exponential functions include:

• $y = 2^x$: This function models the growth of a population over time, where the initial population is 2 and the growth rate is 2.
• $y = 3^x$: This function models the growth of a population over time, where the initial population is 3 and the growth rate is 3.
• $y = 4^x$: This function models the growth of a population over time, where the initial population is 4 and the growth rate is 4.

Solving Exponential Equations

Exponential equations can be solved using logarithms. For example, to solve the equation $2^x = 8$, we can take the logarithm of both sides:

$$\log(2^x) = \log(8)$$

Using the property of logarithms that $\log(a^b) = b \log(a)$, we can simplify this equation to:

$$x \log(2) = \log(8)$$

Dividing both sides by $\log(2)$, we get:

$$x = \frac{\log(8)}{\log(2)}$$

Using a calculator to evaluate the right-hand side, we get:

$$x = 3$$

Therefore, the solution to the equation $2^x = 8$ is $x = 3$.

Conclusion

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Q: What is an exponential function?

A: An exponential function is a function that has the form $y = ab^x$, where $a$ and $b$ are constants and $b > 1$. These functions are used to model a wide range of phenomena, from population growth to chemical reactions.

Q: How do I find the values of $a$ and $b$ given two points on the graph of an exponential function?

A: To find the values of $a$ and $b$, you can use the fact that the function is of the form $y = ab^x$. You can start by using the first point $(1, 1)$ to find the value of $a$. Since $y = ab^x$, you can substitute $x = 1$ and $y = 1$ to get:

$$1 = a \cdot b^1$$

Simplifying this equation, you get:

$$a = \frac{1}{b}$$

Now, you can use the second point $(3, 16)$ to find the value of $b$. Substituting $x = 3$ and $y = 16$ into the equation $y = ab^x$, you get:

$$16 = a \cdot b^3$$

Substituting the expression for $a$ that you found earlier, you get:

$$16 = \frac{1}{b} \cdot b^3$$

Simplifying this equation, you get:

$$16 = b^2$$

Taking the square root of both sides, you get:

$$b = \pm 4$$

Since $b > 1$, you know that $b = 4$.

Q: How do I find the value of $a$ given the value of $b$?

A: To find the value of $a$ given the value of $b$, you can substitute $b = 4$ into the expression $a = \frac{1}{b}$. This gives you:

$$a = \frac{1}{4}$$

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

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

• Population growth: Exponential functions can be used to model the growth of populations over time.
• Chemical reactions: Exponential functions can be used to model the rate of chemical reactions.
• Finance: Exponential functions can be used to model the growth of investments over time.
• Biology: Exponential functions can be used to model the growth of bacteria and other microorganisms.

Q: How do I solve exponential equations?

A: Exponential equations can be solved using logarithms. For example, to solve the equation $2^x = 8$, you can take the logarithm of both sides:

$$\log(2^x) = \log(8)$$

Using the property of logarithms that $\log(a^b) = b \log(a)$, you can simplify this equation to:

$$x \log(2) = \log(8)$$

Dividing both sides by $\log(2)$, you get:

$$x = \frac{\log(8)}{\log(2)}$$

Using a calculator to evaluate the right-hand side, you get:

$$x = 3$$

Therefore, the solution to the equation $2^x = 8$ is $x = 3$.

Q: What are some examples of exponential functions?

A: Some examples of exponential functions include:

• $y = 2^x$: This function models the growth of a population over time, where the initial population is 2 and the growth rate is 2.
• $y = 3^x$: This function models the growth of a population over time, where the initial population is 3 and the growth rate is 3.
• $y = 4^x$: This function models the growth of a population over time, where the initial population is 4 and the growth rate is 4.

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 plot the function. For example, to graph the function $y = 2^x$, you can use the following table of values:

x	y
-2	0.25
-1	0.5
0	1
1	2
2	4

Plotting these points on a coordinate plane, you get a graph of the function $y = 2^x$.

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:

• Not checking the domain of the function: Make sure to check the domain of the function to ensure that it is defined for all values of $x$.
• Not checking the range of the function: Make sure to check the range of the function to ensure that it is defined for all values of $y$.
• Not using the correct formula: Make sure to use the correct formula for the function, such as $y = ab^x$.
• Not checking for extraneous solutions: Make sure to check for extraneous solutions when solving exponential equations.

By avoiding these common mistakes, you can ensure that you are working with exponential functions correctly and accurately.