Evaluate $y=3^x$ When $x=-4$.A. $y=-81$ B. $y=\frac{1}{81}$ C. $y=-12$

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will focus on evaluating the exponential function $y=3^x$ when $x=-4$. We will break down the process step by step and provide a clear explanation of the calculations involved.

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 $y=a^x$, where $a$ is a positive constant and $x$ is the variable. In the case of the function $y=3^x$, $a=3$.

Evaluating Exponential Functions

To evaluate an exponential function, we need to substitute the given value of $x$ into the function and perform the necessary calculations. In this case, we need to find the value of $y$ when $x=-4$.

Step 1: Substitute the Value of $x$

The first step is to substitute the value of $x$ into the function. In this case, we have $x=-4$, so we will replace $x$ with $-4$ in the function $y=3^x$.

y = 3^{-4}

Step 2: Apply the Power Rule

The next step is to apply the power rule, which states that for any positive constant $a$ and any integers $m$ and $n$, $a^{m+n}=a^m \cdot a^n$. In this case, we have $a=3$ and $m=-4$ and $n=0$.

y = 3^{-4} = \frac{1}{3^4}

Step 3: Simplify the Expression

The final step is to simplify the expression by evaluating the exponent. In this case, we have $3^4=81$, so we can rewrite the expression as follows:

y = \frac{1}{81}

Conclusion

In conclusion, to evaluate the exponential function $y=3^x$ when $x=-4$, we need to substitute the value of $x$ into the function, apply the power rule, and simplify the expression. The final answer is $y=\frac{1}{81}$.

Comparison of Options

Let's compare our answer with the options provided:

• Option A: $y=-81$
• Option B: $y=\frac{1}{81}$
• Option C: $y=-12$

Our answer, $y=\frac{1}{81}$, matches option B.

Common Mistakes to Avoid

When evaluating exponential functions, it's essential to avoid common mistakes such as:

• Forgetting to apply the power rule
• Not simplifying the expression correctly
• Confusing the order of operations

By following the steps outlined in this article, you can avoid these mistakes and ensure accurate results.

Real-World Applications

Exponential functions have numerous real-world applications, including:

• Modeling population growth
• Describing chemical reactions
• Analyzing financial data

Understanding how to evaluate exponential functions is crucial for solving problems in these areas.

Final Thoughts

Introduction

In our previous article, we explored the concept of evaluating exponential functions, focusing on the function $y=3^x$ when $x=-4$. We provided a step-by-step guide on how to evaluate this function and compared our answer with the options provided. In this article, we will address some common questions and concerns related to evaluating exponential functions.

Q&A

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

A: An exponential function is a function of the form $y=a^x$, where $a$ is a positive constant and $x$ is the variable. A polynomial function, on the other hand, is a function of the form $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Q: How do I know which base to use when evaluating an exponential function?

A: The base of an exponential function is typically given in the problem statement. If the base is not specified, you can assume it is 10 or $e$ (approximately 2.718), depending on the context.

Q: Can I use a calculator to evaluate an exponential function?

A: Yes, you can use a calculator to evaluate an exponential function. However, it's essential to understand the concept behind the calculation and be able to explain the process.

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

A: An exponential function is a function of the form $y=a^x$, where $a$ is a positive constant and $x$ is the variable. A logarithmic function, on the other hand, is a function of the form $y=\log_a(x)$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I evaluate an exponential function with a negative exponent?

A: To evaluate an exponential function with a negative exponent, you can use the rule $a^{-n}=\frac{1}{a^n}$. For example, $3^{-4}=\frac{1}{3^4}$.

Q: Can I use the power rule to evaluate an exponential function with a negative exponent?

A: Yes, you can use the power rule to evaluate an exponential function with a negative exponent. For example, $3^{-4}=\frac{1}{3^4}=3^{-4}=\frac{1}{81}$.

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

A: An exponential function is a function of the form $y=a^x$, where $a$ is a positive constant and $x$ is the variable. A trigonometric function, on the other hand, is a function of the form $y=\sin(x)$, $y=\cos(x)$, or $y=\tan(x)$, where $x$ is the variable.

Q: How do I evaluate an exponential function with a fractional exponent?

A: To evaluate an exponential function with a fractional exponent, you can use the rule $a^{m/n}=(a^m)^{1/n}$. For example, $3^{2/3}=(3^2)^{1/3}=9^{1/3}=3$.

Q: Can I use the power rule to evaluate an exponential function with a fractional exponent?

A: Yes, you can use the power rule to evaluate an exponential function with a fractional exponent. For example, $3^{2/3}=(3^2)^{1/3}=9^{1/3}=3$.

Conclusion

Evaluating exponential functions is a crucial skill in mathematics, and understanding the concept behind the calculation is essential. By following the steps outlined in this article and addressing common questions and concerns, you can become proficient in evaluating exponential functions and tackle complex mathematical problems with confidence.

Common Mistakes to Avoid

When evaluating exponential functions, it's essential to avoid common mistakes such as:

• Forgetting to apply the power rule
• Not simplifying the expression correctly
• Confusing the order of operations

By following the steps outlined in this article and being aware of these common mistakes, you can ensure accurate results and become proficient in evaluating exponential functions.

Real-World Applications

Exponential functions have numerous real-world applications, including:

• Modeling population growth
• Describing chemical reactions
• Analyzing financial data

Understanding how to evaluate exponential functions is crucial for solving problems in these areas.

Final Thoughts

Evaluating exponential functions may seem daunting at first, but with practice and patience, you can master this skill. Remember to follow the steps outlined in this article, and don't hesitate to ask for help if you're unsure. With time and effort, you'll become proficient in evaluating exponential functions and be able to tackle complex mathematical problems with confidence.