The Equation For An Exponential Function Is $h(x) = 3 \cdot 4^x + 3$. Graph The Function.

Introduction

In mathematics, an exponential function is a function that has the form $f(x) = a \cdot b^x + c$, where $a$, $b$, and $c$ are constants. The equation $h(x) = 3 \cdot 4^x + 3$ is a specific example of an exponential function. In this article, we will graph the function $h(x) = 3 \cdot 4^x + 3$.

Understanding the Equation

The equation $h(x) = 3 \cdot 4^x + 3$ can be broken down into three parts:

• $3 \cdot 4^x$: This is the exponential part of the function, where $4^x$ represents the exponential term. The base of the exponential term is $4$, and the exponent is $x$.
• $3$: This is the constant term of the function, which is added to the exponential term.

Graphing the Function

To graph the function $h(x) = 3 \cdot 4^x + 3$, we can use a graphing calculator or a computer algebra system (CAS). However, we can also graph the function by hand using a table of values.

Table of Values

To create a table of values, we need to choose several values of $x$ and calculate the corresponding values of $h(x)$. Let's choose $x = -2, -1, 0, 1, 2$.

Graphing the Function

Using the table of values, we can graph the function $h(x) = 3 \cdot 4^x + 3$. The graph of the function is a curve that increases rapidly as $x$ increases.

Properties of the Graph

The graph of the function $h(x) = 3 \cdot 4^x + 3$ has several properties:

• Asymptote: The graph of the function has a horizontal asymptote at $y = 3$.
• Domain: The domain of the function is all real numbers, $(-\infty, \infty)$.
• Range: The range of the function is all real numbers greater than or equal to $3$, $[3, \infty)$.

Conclusion

In this article, we graphed the function $h(x) = 3 \cdot 4^x + 3$. We broke down the equation into three parts and created a table of values to graph the function. We also discussed the properties of the graph, including the asymptote, domain, and range.

Exercises

1. Graph the function $f(x) = 2 \cdot 3^x + 2$.
2. Find the equation of the horizontal asymptote of the function $g(x) = 4 \cdot 2^x + 1$.
3. Determine the domain and range of the function $h(x) = 5 \cdot 3^x - 2$.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-20.
• [2] "Graphing Exponential Functions". Purplemath. Retrieved 2023-02-20.
  Q&A: Exponential Functions =============================

Introduction

In our previous article, we graphed the function $h(x) = 3 \cdot 4^x + 3$. In this article, we will answer some frequently asked questions about exponential functions.

Q: What is an exponential function?

A: An exponential function is a function that has the form $f(x) = a \cdot b^x + c$, where $a$, $b$, and $c$ are constants. The base of the exponential term is $b$, and the exponent is $x$.

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

A: An exponential function grows rapidly as $x$ increases, while a linear function grows at a constant rate. For example, the function $f(x) = 2^x$ grows much faster than the function $f(x) = 2x$.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer algebra system (CAS). Alternatively, you can create a table of values and plot the points on a coordinate plane.

Q: What is the horizontal asymptote of an exponential function?

A: The horizontal asymptote of an exponential function is the horizontal line that the function approaches as $x$ increases without bound. For example, the function $f(x) = 2^x$ has a horizontal asymptote at $y = 0$.

Q: What is the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, $(-\infty, \infty)$. The range of an exponential function is all real numbers greater than or equal to the constant term, $[c, \infty)$.

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

A: Yes, exponential functions can be used to model many real-world phenomena, such as population growth, chemical reactions, and financial investments.

Q: How do I determine the equation of an exponential function?

A: To determine the equation of an exponential function, you need to know the base, the exponent, and the constant term. For example, if you know that the function has a base of $2$, an exponent of $x$, and a constant term of $3$, you can write the equation as $f(x) = 3 \cdot 2^x$.

Q: Can I use exponential functions to solve problems involving growth and decay?

A: Yes, exponential functions can be used to solve problems involving growth and decay. For example, if you know that a population is growing at a rate of $2\%$ per year, you can use an exponential function to model the population growth.

Q: How do I use exponential functions to solve problems involving finance?

A: Exponential functions can be used to solve problems involving finance, such as calculating compound interest and present value. For example, if you know that an investment is earning a rate of $5\%$ per year, compounded annually, you can use an exponential function to calculate the future value of the investment.

Conclusion

In this article, we answered some frequently asked questions about exponential functions. We discussed the definition of an exponential function, how to graph an exponential function, and how to use exponential functions to model real-world phenomena.

Exercises

1. Graph the function $f(x) = 2 \cdot 3^x + 2$.
2. Find the equation of the horizontal asymptote of the function $g(x) = 4 \cdot 2^x + 1$.
3. Determine the domain and range of the function $h(x) = 5 \cdot 3^x - 2$.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-20.
• [2] "Graphing Exponential Functions". Purplemath. Retrieved 2023-02-20.