How Many Zero Pairs Must Be Added To The Function $f(x) = X^2 - 10x - 4$ In Order To Begin Writing The Function In Vertex Form?A. 4 B. 10 C. 21 D. 25

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Introduction to Vertex Form

Vertex form is a way of expressing a quadratic function in the form $f(x) = a(x - h)^2 + k$, where (h,k)(h, k) is the vertex of the parabola. To write a quadratic function in vertex form, we need to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial.

Completing the Square

To complete the square for the function $f(x) = x^2 - 10x - 4$, we need to add and subtract a constant term that will make the expression a perfect square trinomial. The constant term we need to add is half of the coefficient of the xx term, squared. In this case, the coefficient of the xx term is −10-10, so we need to add and subtract (−102)2=25(\frac{-10}{2})^2 = 25.

Adding Zero Pairs

When we add and subtract 25, we are essentially adding two zero pairs: 25+0=2525 + 0 = 25 and −25+0=−25-25 + 0 = -25. These zero pairs are added to the function to create a perfect square trinomial.

Writing the Function in Vertex Form

By adding the zero pairs, we can rewrite the function in vertex form:

f(x)=(x−5)2−29f(x) = (x - 5)^2 - 29

Conclusion

To write the function $f(x) = x^2 - 10x - 4$ in vertex form, we need to add 25 to the function. This is equivalent to adding two zero pairs: 25+0=2525 + 0 = 25 and −25+0=−25-25 + 0 = -25. Therefore, the correct answer is:

The final answer is 25.

Step-by-Step Solution

  1. Identify the coefficient of the xx term, which is −10-10.
  2. Calculate the constant term to add by squaring half of the coefficient of the xx term: (−102)2=25(\frac{-10}{2})^2 = 25.
  3. Add and subtract the constant term to create a perfect square trinomial: x2−10x+25−25−4x^2 - 10x + 25 - 25 - 4.
  4. Simplify the expression by combining like terms: (x−5)2−29(x - 5)^2 - 29.
  5. Identify the number of zero pairs added to the function, which is 25.

Frequently Asked Questions

  • What is vertex form? Vertex form is a way of expressing a quadratic function in the form $f(x) = a(x - h)^2 + k$, where (h,k)(h, k) is the vertex of the parabola.
  • What is completing the square? Completing the square is a process of adding and subtracting a constant term to create a perfect square trinomial.
  • How many zero pairs must be added to the function $f(x) = x^2 - 10x - 4$ in order to begin writing the function in vertex form? The correct answer is 25.

Q: What is vertex form?

A: Vertex form is a way of expressing a quadratic function in the form $f(x) = a(x - h)^2 + k$, where (h,k)(h, k) is the vertex of the parabola.

Q: What is completing the square?

A: Completing the square is a process of adding and subtracting a constant term to create a perfect square trinomial. This process involves adding and subtracting a value that will make the expression a perfect square trinomial.

Q: Why do we need to complete the square?

A: We need to complete the square in order to write a quadratic function in vertex form. Vertex form is a useful way to express a quadratic function because it makes it easy to identify the vertex of the parabola.

Q: How do we complete the square?

A: To complete the square, we need to add and subtract a constant term that will make the expression a perfect square trinomial. The constant term we need to add is half of the coefficient of the xx term, squared.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

f(x)=x2+bx+c→f(x)=(x+b2)2−(b2)2+cf(x) = x^2 + bx + c \rightarrow f(x) = (x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c

Q: How do we write a quadratic function in vertex form?

A: To write a quadratic function in vertex form, we need to complete the square. This involves adding and subtracting a constant term that will make the expression a perfect square trinomial.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.

Q: How do we find the vertex of a parabola?

A: To find the vertex of a parabola, we need to write the quadratic function in vertex form. The vertex is then given by the point (h,k)(h, k) in the vertex form.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is significant because it represents the minimum or maximum point of the parabola. It is also the point where the parabola changes direction.

Q: Can we complete the square for any quadratic function?

A: Yes, we can complete the square for any quadratic function. However, the process may be more complicated for some functions.

Q: What are the benefits of completing the square?

A: The benefits of completing the square include:

  • Writing a quadratic function in vertex form
  • Identifying the vertex of a parabola
  • Understanding the behavior of a quadratic function

Q: What are the challenges of completing the square?

A: The challenges of completing the square include:

  • Understanding the process of completing the square
  • Identifying the correct constant term to add
  • Simplifying the expression after completing the square

Q: Can we use technology to complete the square?

A: Yes, we can use technology to complete the square. Many graphing calculators and computer algebra systems can complete the square for us.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

  • Modeling population growth
  • Understanding the behavior of a quadratic function
  • Identifying the vertex of a parabola

Q: Can we use completing the square to solve systems of equations?

A: Yes, we can use completing the square to solve systems of equations. This involves writing the system of equations in vertex form and then solving for the variables.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

  • Not adding and subtracting the correct constant term
  • Not simplifying the expression after completing the square
  • Not identifying the vertex of the parabola

Q: Can we use completing the square to graph a quadratic function?

A: Yes, we can use completing the square to graph a quadratic function. This involves writing the function in vertex form and then graphing the parabola.

Q: What are some tips for completing the square?

A: Some tips for completing the square include:

  • Understanding the process of completing the square
  • Identifying the correct constant term to add
  • Simplifying the expression after completing the square
  • Using technology to check your work

Q: Can we use completing the square to find the equation of a parabola?

A: Yes, we can use completing the square to find the equation of a parabola. This involves writing the equation in vertex form and then identifying the vertex of the parabola.

Q: What are some common applications of completing the square?

A: Some common applications of completing the square include:

  • Modeling population growth
  • Understanding the behavior of a quadratic function
  • Identifying the vertex of a parabola

Q: Can we use completing the square to solve quadratic equations?

A: Yes, we can use completing the square to solve quadratic equations. This involves writing the equation in vertex form and then solving for the variables.

Q: What are some real-world examples of completing the square?

A: Some real-world examples of completing the square include:

  • Modeling the growth of a population
  • Understanding the behavior of a quadratic function
  • Identifying the vertex of a parabola

Q: Can we use completing the square to graph a quadratic function on a calculator?

A: Yes, we can use completing the square to graph a quadratic function on a calculator. This involves writing the function in vertex form and then graphing the parabola.

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

A: Some common mistakes to avoid when graphing a quadratic function include:

  • Not writing the function in vertex form
  • Not identifying the vertex of the parabola
  • Not graphing the parabola correctly

Q: Can we use completing the square to find the equation of a parabola on a calculator?

A: Yes, we can use completing the square to find the equation of a parabola on a calculator. This involves writing the equation in vertex form and then identifying the vertex of the parabola.

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

A: Some real-world applications of graphing a quadratic function include:

  • Modeling population growth
  • Understanding the behavior of a quadratic function
  • Identifying the vertex of a parabola

Q: Can we use completing the square to solve systems of equations on a calculator?

A: Yes, we can use completing the square to solve systems of equations on a calculator. This involves writing the system of equations in vertex form and then solving for the variables.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

  • Not writing the system of equations in vertex form
  • Not identifying the vertex of the parabola
  • Not solving for the variables correctly

Q: Can we use completing the square to find the equation of a parabola on a graphing calculator?

A: Yes, we can use completing the square to find the equation of a parabola on a graphing calculator. This involves writing the equation in vertex form and then identifying the vertex of the parabola.

Q: What are some real-world applications of finding the equation of a parabola?

A: Some real-world applications of finding the equation of a parabola include:

  • Modeling population growth
  • Understanding the behavior of a quadratic function
  • Identifying the vertex of a parabola