Given The Function \[$ G(x) = 5x - 12x^2 + 3 \$\], Find The Value Of \[$ G(x) \$\] For A Specified \[$ X \$\].
Introduction
In mathematics, polynomial functions are a fundamental concept in algebra and calculus. A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. In this article, we will focus on evaluating a given polynomial function, { g(x) = 5x - 12x^2 + 3 $}$, for a specified value of { x $}$.
Understanding Polynomial Functions
A polynomial function is typically denoted as { p(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 the variable { x $}$ in the function.
Evaluating the Given Polynomial Function
To evaluate the given polynomial function, { g(x) = 5x - 12x^2 + 3 $}$, for a specified value of { x $}$, we need to substitute the value of { x $}$ into the function and simplify the expression.
Step 1: Substitute the Value of { x $}$
Let's say we want to evaluate the function at { x = 2 $}$. We will substitute { x = 2 $}$ into the function:
{ g(2) = 5(2) - 12(2)^2 + 3 $}$
Step 2: Simplify the Expression
Now, we will simplify the expression by evaluating the products and combining like terms:
{ g(2) = 10 - 12(4) + 3 $}$
{ g(2) = 10 - 48 + 3 $}$
{ g(2) = -35 $}$
Therefore, the value of the function { g(x) $}$ at { x = 2 $}$ is { -35 $}$.
Example 2: Evaluating the Function at { x = -3 $}$
Let's say we want to evaluate the function at { x = -3 $}$. We will substitute { x = -3 $}$ into the function:
{ g(-3) = 5(-3) - 12(-3)^2 + 3 $}$
{ g(-3) = -15 - 12(9) + 3 $}$
{ g(-3) = -15 - 108 + 3 $}$
{ g(-3) = -120 $}$
Therefore, the value of the function { g(x) $}$ at { x = -3 $}$ is { -120 $}$.
Conclusion
In this article, we have discussed how to evaluate a given polynomial function, { g(x) = 5x - 12x^2 + 3 $}$, for a specified value of { x $}$. We have provided step-by-step examples of how to substitute the value of { x $}$ into the function and simplify the expression. By following these steps, you can evaluate any polynomial function for a specified value of { x $}$.
Tips and Tricks
- Make sure to substitute the value of { x $}$ into the function correctly.
- Simplify the expression by evaluating the products and combining like terms.
- Check your work by plugging the value of { x $}$ back into the function to ensure that the result is correct.
Common Mistakes to Avoid
- Failing to substitute the value of { x $}$ into the function correctly.
- Not simplifying the expression by evaluating the products and combining like terms.
- Not checking the work by plugging the value of { x $}$ back into the function to ensure that the result is correct.
Real-World Applications
Polynomial functions have numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile can be modeled using a polynomial function, and the cost of producing a product can be represented by a polynomial function.
Final Thoughts
Introduction
In our previous article, we discussed how to evaluate a given polynomial function, { g(x) = 5x - 12x^2 + 3 $}$, for a specified value of { x $}$. In this article, we will provide a Q&A guide to help you better understand the concept of evaluating polynomial functions.
Q&A
Q: What is a polynomial function?
A: A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.
Q: What is the degree of a polynomial function?
A: The degree of a polynomial function is the highest power of the variable { x $}$ in the function.
Q: How do I evaluate a polynomial function for a specified value of { x $}$?
A: To evaluate a polynomial function for a specified value of { x $}$, you need to substitute the value of { x $}$ into the function and simplify the expression.
Q: What is the difference between evaluating a polynomial function and finding the derivative of a polynomial function?
A: Evaluating a polynomial function involves substituting a value of { x $}$ into the function and simplifying the expression, whereas finding the derivative of a polynomial function involves finding the rate of change of the function with respect to { x $}$.
Q: Can I use a calculator to evaluate a polynomial function?
A: Yes, you can use a calculator to evaluate a polynomial function. However, it's always a good idea to check your work by plugging the value of { x $}$ back into the function to ensure that the result is correct.
Q: What are some common mistakes to avoid when evaluating a polynomial function?
A: Some common mistakes to avoid when evaluating a polynomial function include failing to substitute the value of { x $}$ into the function correctly, not simplifying the expression by evaluating the products and combining like terms, and not checking the work by plugging the value of { x $}$ back into the function to ensure that the result is correct.
Q: How do I know if a polynomial function is a good model for a real-world problem?
A: To determine if a polynomial function is a good model for a real-world problem, you need to consider the following factors:
- Does the function accurately represent the data?
- Is the function consistent with the laws of physics and mathematics?
- Does the function make sense in the context of the problem?
Q: Can I use a polynomial function to model a non-linear relationship?
A: Yes, you can use a polynomial function to model a non-linear relationship. However, the degree of the polynomial function will depend on the complexity of the relationship.
Q: How do I determine the degree of a polynomial function?
A: To determine the degree of a polynomial function, you need to identify the highest power of the variable { x $}$ in the function.
Q: Can I use a polynomial function to model a periodic relationship?
A: Yes, you can use a polynomial function to model a periodic relationship. However, the degree of the polynomial function will depend on the complexity of the relationship.
Q: How do I determine if a polynomial function is a good model for a periodic relationship?
A: To determine if a polynomial function is a good model for a periodic relationship, you need to consider the following factors:
- Does the function accurately represent the data?
- Is the function consistent with the laws of physics and mathematics?
- Does the function make sense in the context of the problem?
Conclusion
Evaluating polynomial functions is an essential skill in mathematics and has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can evaluate any polynomial function for a specified value of { x $}$. Remember to substitute the value of { x $}$ into the function correctly, simplify the expression by evaluating the products and combining like terms, and check your work by plugging the value of { x $}$ back into the function to ensure that the result is correct.
Tips and Tricks
- Make sure to substitute the value of { x $}$ into the function correctly.
- Simplify the expression by evaluating the products and combining like terms.
- Check your work by plugging the value of { x $}$ back into the function to ensure that the result is correct.
- Use a calculator to evaluate a polynomial function, but always check your work.
- Consider the factors mentioned above when determining if a polynomial function is a good model for a real-world problem.
Real-World Applications
Polynomial functions have numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile can be modeled using a polynomial function, and the cost of producing a product can be represented by a polynomial function.
Final Thoughts
Evaluating polynomial functions is an essential skill in mathematics and has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can evaluate any polynomial function for a specified value of { x $}$. Remember to substitute the value of { x $}$ into the function correctly, simplify the expression by evaluating the products and combining like terms, and check your work by plugging the value of { x $}$ back into the function to ensure that the result is correct.