Evaluate $f(x) = 2x^2 + 1$ At $x = -2$.

Introduction

In mathematics, evaluating a function at a specific value of its variable is a fundamental concept. It involves substituting the given value into the function and simplifying the expression to obtain the result. In this article, we will evaluate the quadratic function $f(x) = 2x^2 + 1$ at $x = -2$. This will help us understand the process of evaluating a quadratic function and how to apply it in real-world problems.

Understanding the Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = 2$, $b = 0$, and $c = 1$. The function represents a parabola that opens upwards, since the coefficient of $x^2$ is positive.

Evaluating the Function at $x = -2$

To evaluate the function at $x = -2$, we need to substitute $x = -2$ into the function and simplify the expression. This can be done using the following steps:

1. Substitute $x = -2$ into the function: Replace $x$ with $-2$ in the function $f(x) = 2x^2 + 1$.
2. Simplify the expression: Expand and simplify the resulting expression to obtain the final result.

Step 1: Substitute $x = -2$ into the function

$f(-2) = 2(-2)^2 + 1$

Step 2: Simplify the expression

To simplify the expression, we need to expand the squared term and then combine like terms.

$f(-2) = 2(-4) + 1$

$f(-2) = -8 + 1$

$f(-2) = -7$

Conclusion

In this article, we evaluated the quadratic function $f(x) = 2x^2 + 1$ at $x = -2$. We followed the steps of substituting $x = -2$ into the function and simplifying the resulting expression to obtain the final result. The process of evaluating a quadratic function is an essential concept in mathematics, and it has numerous applications in real-world problems.

Real-World Applications

Evaluating a quadratic function has numerous applications in real-world problems, such as:

• Physics: To calculate the trajectory of an object under the influence of gravity.
• Engineering: To design and optimize systems, such as bridges and buildings.
• Economics: To model and analyze economic systems, such as supply and demand.

Tips and Tricks

When evaluating a quadratic function, it's essential to follow the steps carefully and simplify the expression correctly. Here are some tips and tricks to help you evaluate quadratic functions:

• Use the correct order of operations: When simplifying the expression, use the correct order of operations (PEMDAS) to avoid errors.
• Combine like terms: Combine like terms to simplify the expression and obtain the final result.
• Check your work: Double-check your work to ensure that you have obtained the correct result.

Common Mistakes

When evaluating a quadratic function, it's easy to make mistakes. Here are some common mistakes to avoid:

• Incorrect substitution: Make sure to substitute the correct value of $x$ into the function.
• Incorrect simplification: Simplify the expression correctly using the correct order of operations.
• Incorrect result: Double-check your work to ensure that you have obtained the correct result.

Conclusion

Introduction

In our previous article, we evaluated the quadratic function $f(x) = 2x^2 + 1$ at $x = -2$. In this article, we will answer some frequently asked questions about evaluating quadratic functions.

Q: What is a quadratic function?

A quadratic function is a polynomial function of degree two, which means it has the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I evaluate a quadratic function?

To evaluate a quadratic function, you need to substitute the given value of $x$ into the function and simplify the expression. This can be done using the following steps:

1. Substitute $x$ into the function: Replace $x$ with the given value in the function.
2. Simplify the expression: Expand and simplify the resulting expression to obtain the final result.

Q: What is the difference between evaluating a quadratic function and finding its roots?

Evaluating a quadratic function involves substituting a given value of $x$ into the function and simplifying the expression to obtain the result. Finding the roots of a quadratic function involves solving the equation $f(x) = 0$ to obtain the values of $x$ that make the function equal to zero.

Q: How do I find the roots of a quadratic function?

To find the roots of a quadratic function, you need to solve the equation $f(x) = 0$. This can be done using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: What is the quadratic formula?

The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

To use the quadratic formula, you need to substitute the values of $a$, $b$, and $c$ into the formula and simplify the expression to obtain the roots of the equation.

Q: What are some common mistakes to avoid when evaluating a quadratic function?

Some common mistakes to avoid when evaluating a quadratic function include:

• Incorrect substitution: Make sure to substitute the correct value of $x$ into the function.
• Incorrect simplification: Simplify the expression correctly using the correct order of operations.
• Incorrect result: Double-check your work to ensure that you have obtained the correct result.

Q: How do I check my work when evaluating a quadratic function?

To check your work when evaluating a quadratic function, you need to:

• Re-evaluate the expression: Re-evaluate the expression to ensure that you have obtained the correct result.
• Use a calculator: Use a calculator to check your work and ensure that you have obtained the correct result.
• Check for errors: Check for errors in your work and correct them if necessary.

Conclusion

Evaluating a quadratic function is an essential concept in mathematics, and it has numerous applications in real-world problems. By following the steps carefully and simplifying the expression correctly, you can obtain the final result and apply it in real-world problems. Remember to use the correct order of operations, combine like terms, and check your work to avoid errors.

Additional Resources

For more information on evaluating quadratic functions, you can refer to the following resources:

• Textbooks: Refer to your textbook for more information on evaluating quadratic functions.
• Online resources: Refer to online resources such as Khan Academy, Mathway, and Wolfram Alpha for more information on evaluating quadratic functions.
• Practice problems: Practice evaluating quadratic functions using practice problems to reinforce your understanding of the concept.