Given $f(x) = -3x^2 - 3x$, Find $f(-6$\]. Answer Attempt 1 Out Of 2: $\square$

Introduction

Polynomial functions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will explore how to evaluate a polynomial function at a given value, using the function $f(x) = -3x^2 - 3x$ as an example. We will also provide a step-by-step guide on how to find $f(-6)$.

Understanding Polynomial Functions

A polynomial function is a function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a non-negative integer. The degree of a polynomial function is the highest power of $x$ in the function.

Evaluating Polynomial Functions

To evaluate a polynomial function at a given value, we simply substitute the value into the function and simplify. Let's consider the function $f(x) = -3x^2 - 3x$ and evaluate it at $x = -6$.

Step 1: Substitute the Value into the Function

To evaluate $f(-6)$, we substitute $x = -6$ into the function $f(x) = -3x^2 - 3x$. This gives us:

$f(-6) = -3(-6)^2 - 3(-6)$

Step 2: Simplify the Expression

Now, we simplify the expression by evaluating the exponent and multiplying the terms.

$f(-6) = -3(36) + 18$

$f(-6) = -108 + 18$

$f(-6) = -90$

Conclusion

In this article, we have shown how to evaluate a polynomial function at a given value, using the function $f(x) = -3x^2 - 3x$ as an example. We have provided a step-by-step guide on how to find $f(-6)$, and we have demonstrated how to simplify the expression to obtain the final answer.

Tips and Tricks

• When evaluating a polynomial function, always substitute the value into the function and simplify.
• Use the order of operations (PEMDAS) to simplify the expression.
• Make sure to evaluate the exponent before multiplying the terms.

Common Mistakes

• Failing to substitute the value into the function.
• Not simplifying the expression correctly.
• Not following the order of operations (PEMDAS).

Real-World Applications

Polynomial functions have many real-world applications, including:

• Modeling population growth and decline.
• Describing the motion of objects.
• Analyzing data and making predictions.

Conclusion

Introduction

In our previous article, we explored how to evaluate a polynomial function at a given value, using the function $f(x) = -3x^2 - 3x$ as an example. In this article, we will provide a Q&A guide to help you better understand how to evaluate polynomial functions.

Q&A

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a non-negative integer.

Q: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of $x$ in the function.

Q: How do I evaluate a polynomial function at a given value?

A: To evaluate a polynomial function at a given value, you simply substitute the value into the function and simplify.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next.
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression using the order of operations (PEMDAS)?

A: To simplify an expression using the order of operations (PEMDAS), follow these steps:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponential expressions.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: What are some common mistakes to avoid when evaluating polynomial functions?

A: Some common mistakes to avoid when evaluating polynomial functions include:

• Failing to substitute the value into the function.
• Not simplifying the expression correctly.
• Not following the order of operations (PEMDAS).

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

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

• Modeling population growth and decline.
• Describing the motion of objects.
• Analyzing data and making predictions.

Conclusion

In conclusion, evaluating polynomial functions is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can confidently evaluate polynomial functions and make predictions in a variety of real-world applications.

Additional Resources

• For more information on polynomial functions, visit the Khan Academy website.
• For practice problems and exercises, visit the Mathway website.
• For a comprehensive guide to evaluating polynomial functions, visit the Wolfram MathWorld website.

Final Tips

• Practice, practice, practice! The more you practice evaluating polynomial functions, the more confident you will become.
• Use online resources, such as Khan Academy and Mathway, to supplement your learning.
• Don't be afraid to ask for help if you are struggling with a particular concept or problem.