Write The Exponential Function Y = A B X Y = Ab^x Y = A B X That Contains The Given Points:1. (1, 10) And (2, 20)2. (-1, 4.5) And (2, 36)

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 exponential functions that contain given points. We will explore two different scenarios and provide step-by-step solutions to each problem.

Scenario 1: Writing an Exponential Function with Two Points

Problem

Write the exponential function $y = ab^x$ that contains the given points: (1, 10) and (2, 20).

Solution

To write the exponential function, we need to find the values of $a$ and $b$. We can use the given points to create a system of equations.

Let's start by substituting the first point (1, 10) into the equation:

$$10 = ab^1$$

Simplifying the equation, we get:

$$10 = ab$$

Now, let's substitute the second point (2, 20) into the equation:

$$20 = ab^2$$

We can divide the second equation by the first equation to eliminate $a$:

$$\frac{20}{10} = \frac{ab^2}{ab}$$

Simplifying the equation, we get:

$$2 = b$$

Now that we have found the value of $b$, we can substitute it back into one of the original equations to find the value of $a$. Let's use the first equation:

$$10 = ab$$

Substituting $b = 2$, we get:

$$10 = 2a$$

Dividing both sides by 2, we get:

$$a = 5$$

Therefore, the exponential function that contains the given points is:

$$y = 5(2)^x$$

Verification

Let's verify that the function satisfies the given points:

For the point (1, 10):

$$y = 5(2)^1 = 10$$

For the point (2, 20):

$$y = 5(2)^2 = 20$$

The function satisfies both points, so we have found the correct exponential function.

Scenario 2: Writing an Exponential Function with Two Points

Problem

Write the exponential function $y = ab^x$ that contains the given points: (-1, 4.5) and (2, 36).

Solution

To write the exponential function, we need to find the values of $a$ and $b$. We can use the given points to create a system of equations.

Let's start by substituting the first point (-1, 4.5) into the equation:

$$4.5 = ab^{-1}$$

Simplifying the equation, we get:

$$4.5 = \frac{a}{b}$$

Now, let's substitute the second point (2, 36) into the equation:

$$36 = ab^2$$

We can multiply the first equation by $b$ to eliminate $a$:

$$4.5b = a$$

Now, substitute this expression for $a$ into the second equation:

$$36 = (4.5b)b^2$$

Simplifying the equation, we get:

$$36 = 4.5b^3$$

Dividing both sides by 4.5, we get:

$$8 = b^3$$

Taking the cube root of both sides, we get:

$$b = 2$$

Now that we have found the value of $b$, we can substitute it back into one of the original equations to find the value of $a$. Let's use the first equation:

$$4.5 = \frac{a}{b}$$

Substituting $b = 2$, we get:

$$4.5 = \frac{a}{2}$$

Multiplying both sides by 2, we get:

$$a = 9$$

Therefore, the exponential function that contains the given points is:

$$y = 9(2)^x$$

Verification

Let's verify that the function satisfies the given points:

For the point (-1, 4.5):

$$y = 9(2)^{-1} = 4.5$$

For the point (2, 36):

$$y = 9(2)^2 = 36$$

The function satisfies both points, so we have found the correct exponential function.

Conclusion

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Q&A: Frequently Asked Questions about Exponential Functions

Q: What is an exponential function?

A: An exponential function is a mathematical function that describes a relationship between two variables, where the dependent variable (y) is a constant raised to a power of the independent variable (x). The general form of an exponential function is:

$$y = ab^x$$

where $a$ and $b$ are constants.

Q: How do I write an exponential function that contains a given point?

A: To write an exponential function that contains a given point, you need to find the values of $a$ and $b$. You can use the given point to create a system of equations. Let's say you have a point $(x_1, y_1)$. You can substitute this point into the equation:

$$y_1 = ab^{x_1}$$

You can also use another point $(x_2, y_2)$ to create another equation:

$$y_2 = ab^{x_2}$$

You can then solve the system of equations to find the values of $a$ and $b$.

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

A: An exponential function and a linear function are two different types of mathematical functions. A linear function is a function that can be written in the form:

$$y = mx + b$$

where $m$ and $b$ are constants. An exponential function, on the other hand, is a function that can be written in the form:

$$y = ab^x$$

where $a$ and $b$ are constants. The key difference between the two functions is that a linear function has a constant rate of change, while an exponential function has a rate of change that changes over time.

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. Let's say you have an exponential function:

$$y = 2^x$$

You can create a table of values by plugging in different values of $x$ and calculating the corresponding values of $y$. You can then plot the points on a coordinate plane to create a graph.

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

A: Exponential functions have many real-world applications. Some examples include:

• 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 calculate compound interest, where the interest is added to the principal at regular intervals.
• Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases over time.
• Epidemics: Exponential functions can be used to model the spread of diseases, where the number of infected individuals grows at a constant rate.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable. Let's say you have an exponential equation:

$$2^x = 8$$

You can solve for $x$ by using the fact that $2^3 = 8$. You can then take the logarithm of both sides to get:

$$x = 3$$

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

A: An exponential function and a logarithmic function are two different types of mathematical functions. An exponential function is a function that can be written in the form:

$$y = ab^x$$

where $a$ and $b$ are constants. A logarithmic function, on the other hand, is a function that can be written in the form:

$$y = \log_b x$$

where $b$ is a constant. The key difference between the two functions is that an exponential function grows rapidly, while a logarithmic function grows slowly.

Conclusion

In this article, we have answered some frequently asked questions about exponential functions. We have covered topics such as writing exponential functions, graphing exponential functions, and solving exponential equations. We hope that this article has provided a comprehensive guide to exponential functions and has helped you to understand this important concept in mathematics.