Determining the Number of Zero Pairs

Oct 13, 2025 by windailys.com 75 views

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Introduction

In algebra, quadratic functions are often represented in various forms, including standard form, vertex form, and factored form. The vertex form of a quadratic function is particularly useful for graphing and analyzing the function's behavior. In this article, we will explore how to convert a quadratic function from standard form to vertex form and determine the number of zero pairs that must be added to the function.

Understanding Vertex Form

The vertex form of a quadratic function is represented as:

f(x) = a(x - h)^2 + k

where $(h, k)$ is the vertex of the parabola, and $a$ is the coefficient of the squared term. To convert a quadratic function from standard form to vertex form, we need to complete the square.

Completing the Square

Completing the square involves adding and subtracting a constant term to create a perfect square trinomial. For the given function $$f(x) = x^2 - 10x - 4$$, we can start by adding and subtracting the square of half the coefficient of the linear term.

Step 1: Find the Constant Term to Add

The coefficient of the linear term is $-10$ . Half of this value is $-5$ , and the square of $-5$ is $25$ . We will add and subtract $25$ to the function.

Step 2: Rewrite the Function

f(x) = x^2 - 10x + 25 - 25 - 4

Step 3: Factor the Perfect Square Trinomial

f(x) = (x - 5)^2 - 29

Step 4: Identify the Vertex

The vertex of the parabola is $(h, k) = (5, -29)$ .

Determining the Number of Zero Pairs

To determine the number of zero pairs that must be added to the function, we need to consider the constant term that was added to complete the square. In this case, we added $25$ to the function. Since we added a positive constant term, we need to add a zero pair to the function to maintain the same value.

Conclusion

In conclusion, to convert the quadratic function $$f(x) = x^2 - 10x - 4$$ to vertex form, we need to add a zero pair to the function. The correct answer is:

A. 4

This is because we added a positive constant term of $25$ to the function, and we need to add a zero pair to maintain the same value.

Final Answer

The final answer is $\boxed{4}$ .

Additional Information

Vertex Form: The vertex form of a quadratic function is represented as $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola, and $a$ is the coefficient of the squared term.
Completing the Square: Completing the square involves adding and subtracting a constant term to create a perfect square trinomial.
Zero Pairs: A zero pair is a pair of terms that add up to zero. In the context of completing the square, a zero pair is added to the function to maintain the same value.

References

[1] Algebra.com. (n.d.). Completing the Square. Retrieved from https://www.algebra.com/algebra/homework/Completing-the-Square/Completing-the-Square.html
[2] Khan Academy. (n.d.). Completing the square. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4d7/x2f6f4d7

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

A: The vertex form of a quadratic function is represented as $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola, and $a$ is the coefficient of the squared term.

Q: How do I convert a quadratic function from standard form to vertex form?

A: To convert a quadratic function from standard form to vertex form, you need to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: What is completing the square?

A: Completing the square involves adding and subtracting a constant term to create a perfect square trinomial. This allows you to rewrite the quadratic function in vertex form.

Q: How do I find the constant term to add when completing the square?

A: To find the constant term to add, you need to take half of the coefficient of the linear term and square it. This will give you the constant term to add.

Q: What is a zero pair?

A: A zero pair is a pair of terms that add up to zero. In the context of completing the square, a zero pair is added to the function to maintain the same value.

Q: How many zero pairs must be added to the function to begin writing the function in vertex form?

A: The number of zero pairs that must be added to the function depends on the constant term that was added to complete the square. If a positive constant term was added, you need to add a zero pair to the function to maintain the same value.

Q: What is the final answer to the problem of how many zero pairs must be added to the function?

A: The final answer is $\boxed{4}$ .

Q: What are some related topics to quadratic functions and vertex form?

A: Some related topics to quadratic functions and vertex form include:

Graphing Quadratic Functions: Graphing quadratic functions involves plotting the function on a coordinate plane and identifying key features such as the vertex, axis of symmetry, and x-intercepts.
Analyzing Quadratic Functions: Analyzing quadratic functions involves identifying key features such as the vertex, axis of symmetry, and x-intercepts, and using this information to make predictions about the function's behavior.
Quadratic Equations: Quadratic equations are polynomial equations of degree two. They can be solved using various methods, including factoring, the quadratic formula, and completing the square.

Q: Where can I find more information on quadratic functions and vertex form?

A: You can find more information on quadratic functions and vertex form on various online resources, including:

Algebra.com: Algebra.com is a website that provides interactive algebra lessons and exercises.
Khan Academy: Khan Academy is a website that provides free online courses and resources on a variety of subjects, including algebra and mathematics.
Mathway: Mathway is a website that provides step-by-step solutions to math problems, including quadratic equations and functions.

Q: What are some common mistakes to avoid when converting quadratic functions to vertex form?

A: Some common mistakes to avoid when converting quadratic functions to vertex form include:

Not completing the square correctly: Make sure to add and subtract the correct constant term to create a perfect square trinomial.
Not identifying the vertex correctly: Make sure to identify the vertex of the parabola correctly, including the x-coordinate and y-coordinate.
Not using the correct formula: Make sure to use the correct formula for the vertex form of a quadratic function, which is $f(x) = a(x - h)^2 + k$ .