When Simplified And Written In Standard Form, Which Quadratic Function Is Equivalent To The Polynomial Shown?Options:A. − 4 C 2 + 4 C + 6 -4c^2 + 4c + 6 − 4 C 2 + 4 C + 6 B. − 7 C 2 + 10 C + 6 -7c^2 + 10c + 6 − 7 C 2 + 10 C + 6 C. − 4 C 2 + 4 C + 8 -4c^2 + 4c + 8 − 4 C 2 + 4 C + 8 D. − 7 C 2 + 7 C + 6 -7c^2 + 7c + 6 − 7 C 2 + 7 C + 6 Given

Introduction

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will explore the process of simplifying quadratic functions, with a focus on identifying the equivalent polynomial form. We will use a step-by-step approach to simplify the given polynomial and determine which quadratic function is equivalent to it.

Understanding Quadratic Functions

A quadratic function is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic function is:

f(x) = ax^2 + bx + c

where a, b, and c are constants, and x is the variable. The graph of a quadratic function is a parabola, which can be either upward-facing or downward-facing, depending on the value of a.

Simplifying the Given Polynomial

The given polynomial is:

c^2 - 4c + 6

To simplify this polynomial, we need to rewrite it in standard form, which is:

f(c) = -4c^2 + 4c + 6

Step 1: Expand the Polynomial

The given polynomial is already in its simplest form, so we don't need to expand it.

Step 2: Combine Like Terms

There are no like terms in the given polynomial, so we don't need to combine them.

Step 3: Rewrite the Polynomial in Standard Form

The given polynomial is already in standard form, so we don't need to rewrite it.

Comparing the Simplified Polynomial with the Options

Now that we have simplified the given polynomial, we can compare it with the options provided:

A. $-4c^2 + 4c + 6$ B. $-7c^2 + 10c + 6$ C. $-4c^2 + 4c + 8$ D. $-7c^2 + 7c + 6$

Conclusion

Based on our analysis, we can see that the simplified polynomial is equivalent to option A: $-4c^2 + 4c + 6$. This is because the coefficients of the terms in the simplified polynomial match the coefficients in option A.

Final Answer

The final answer is option A: $-4c^2 + 4c + 6$.

Additional Tips and Tricks

• When simplifying quadratic functions, make sure to rewrite them in standard form, which is ax^2 + bx + c.
• Use the distributive property to expand the polynomial, if necessary.
• Combine like terms to simplify the polynomial.
• Compare the simplified polynomial with the options provided to determine which one is equivalent.

Common Mistakes to Avoid

• Failing to rewrite the polynomial in standard form.
• Not using the distributive property to expand the polynomial.
• Not combining like terms to simplify the polynomial.
• Not comparing the simplified polynomial with the options provided.

Real-World Applications

Quadratic functions have numerous real-world applications, including:

• Modeling the trajectory of a projectile.
• Describing the motion of an object under the influence of gravity.
• Analyzing the behavior of a population growth model.
• Solving optimization problems.

Conclusion

Introduction

In our previous article, we explored the process of simplifying quadratic functions, with a focus on identifying the equivalent polynomial form. In this article, we will answer some frequently asked questions (FAQs) related to quadratic function simplification.

Q: What is the general form of a quadratic function?

A: The general form of a quadratic function is:

f(x) = ax^2 + bx + c

where a, b, and c are constants, and x is the variable.

Q: How do I simplify a quadratic function?

A: To simplify a quadratic function, follow these steps:

1. Rewrite the polynomial in standard form, which is ax^2 + bx + c.
2. Use the distributive property to expand the polynomial, if necessary.
3. Combine like terms to simplify the polynomial.
4. Compare the simplified polynomial with the options provided to determine which one is equivalent.

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

A: A quadratic function is a polynomial of degree two, which means it has a highest power of two. A polynomial, on the other hand, is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can I simplify a quadratic function with a negative leading coefficient?

A: Yes, you can simplify a quadratic function with a negative leading coefficient. The process is the same as for a quadratic function with a positive leading coefficient.

Q: How do I determine the equivalent polynomial form of a quadratic function?

A: To determine the equivalent polynomial form of a quadratic function, compare the simplified polynomial with the options provided. The equivalent polynomial form is the one that matches the coefficients of the terms in the simplified polynomial.

Q: Can I use a calculator to simplify a quadratic function?

A: Yes, you can use a calculator to simplify a quadratic function. However, it's essential to understand the process of simplifying a quadratic function manually to ensure accuracy and to develop problem-solving skills.

Q: What are some real-world applications of quadratic function simplification?

A: Quadratic function simplification has numerous real-world applications, including:

• Modeling the trajectory of a projectile.
• Describing the motion of an object under the influence of gravity.
• Analyzing the behavior of a population growth model.
• Solving optimization problems.

Q: Can I simplify a quadratic function with a variable in the denominator?

A: No, you cannot simplify a quadratic function with a variable in the denominator. The process of simplifying a quadratic function requires a polynomial with a variable in the numerator and a constant in the denominator.

Q: How do I know if a quadratic function is equivalent to a given polynomial?

A: To determine if a quadratic function is equivalent to a given polynomial, compare the coefficients of the terms in the quadratic function with the coefficients of the terms in the given polynomial. If they match, then the quadratic function is equivalent to the given polynomial.

Conclusion

In conclusion, quadratic function simplification is an essential skill in mathematics, and it requires a step-by-step approach. By following the steps outlined in this article and answering the FAQs, you can simplify quadratic functions and determine which one is equivalent to a given polynomial. Remember to rewrite the polynomial in standard form, use the distributive property to expand it, combine like terms to simplify it, and compare the simplified polynomial with the options provided.