What Is The First Step When Rewriting Y = − 4 X 2 + 2 X − 7 Y=-4x^2+2x-7 Y = − 4 X 2 + 2 X − 7 In The Form Y = A ( X − H ) 2 + K Y=a(x-h)^2+k Y = A ( X − H ) 2 + K ?A. 2 Must Be Factored From 2 X − 7 2x-7 2 X − 7 .B. -4 Must Be Factored From − 4 X 2 + 2 X -4x^2+2x − 4 X 2 + 2 X .C. X X X Must Be Factored From − 4 X 2 + 2 X -4x^2+2x − 4 X 2 + 2 X .D. -4

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What is the First Step When Rewriting y=−4x2+2x−7y=-4x^2+2x-7 in the Form y=a(x−h)2+ky=a(x-h)^2+k?

Understanding the Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is given by y=a(x−h)2+ky=a(x-h)^2+k, where (h,k)(h,k) represents the coordinates of the vertex of the parabola. This form is particularly useful for identifying the maximum or minimum value of a quadratic function. To rewrite a quadratic equation in vertex form, we need to complete the square, which involves manipulating the equation to express it in the desired form.

Step 1: Identify the Coefficient of the x2x^2 Term

The first step in rewriting the quadratic equation y=−4x2+2x−7y=-4x^2+2x-7 in vertex form is to identify the coefficient of the x2x^2 term, which is −4-4. This coefficient will be used to determine the value of aa in the vertex form.

Step 2: Factor Out the Coefficient of the x2x^2 Term

To factor out the coefficient of the x2x^2 term, we need to group the terms that have the same power of xx. In this case, we can group the x2x^2 term and the xx term together. This gives us:

y=−4(x2−12x)−7y=-4(x^2-\frac{1}{2}x)-7

Step 3: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of the xx term inside the parentheses. In this case, the coefficient of the xx term is −12-\frac{1}{2}, so we need to add and subtract (14)2=116\left(\frac{1}{4}\right)^2=\frac{1}{16} inside the parentheses.

y=−4(x2−12x+116)−7+416y=-4(x^2-\frac{1}{2}x+\frac{1}{16})-7+\frac{4}{16}

Step 4: Simplify the Expression

Now that we have completed the square, we can simplify the expression by combining like terms.

y=−4(x−18)2−554y=-4(x-\frac{1}{8})^2-\frac{55}{4}

Conclusion

In conclusion, the first step when rewriting the quadratic equation y=−4x2+2x−7y=-4x^2+2x-7 in vertex form is to identify the coefficient of the x2x^2 term, which is −4-4. We then factor out this coefficient to group the terms that have the same power of xx. Finally, we complete the square by adding and subtracting the square of half the coefficient of the xx term inside the parentheses.

Answer

The correct answer is B. -4 must be factored from −4x2+2x-4x^2+2x.

Discussion

The vertex form of a quadratic equation is a powerful tool for identifying the maximum or minimum value of a quadratic function. By completing the square, we can rewrite a quadratic equation in vertex form, which allows us to easily identify the coordinates of the vertex. In this case, we factored out the coefficient of the x2x^2 term, which is −4-4, to group the terms that have the same power of xx. This allowed us to complete the square and rewrite the quadratic equation in vertex form.

Additional Tips

  • When rewriting a quadratic equation in vertex form, it's essential to identify the coefficient of the x2x^2 term and factor it out to group the terms that have the same power of xx.
  • Completing the square involves adding and subtracting the square of half the coefficient of the xx term inside the parentheses.
  • The vertex form of a quadratic equation is given by y=a(x−h)2+ky=a(x-h)^2+k, where (h,k)(h,k) represents the coordinates of the vertex of the parabola.

Related Topics

  • Completing the square
  • Vertex form of a quadratic equation
  • Quadratic functions
  • Maximum and minimum values of a quadratic function

References

  • [1] "Completing the Square" by Math Open Reference
  • [2] "Vertex Form of a Quadratic Equation" by Purplemath
  • [3] "Quadratic Functions" by Khan Academy
    Q&A: Completing the Square and Vertex Form of a Quadratic Equation

Frequently Asked Questions

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic equation in vertex form. It involves adding and subtracting a constant term inside the parentheses to create a perfect square trinomial.

Q: Why is completing the square important?

A: Completing the square is important because it allows us to rewrite a quadratic equation in vertex form, which makes it easier to identify the maximum or minimum value of the function. It also helps us to find the coordinates of the vertex of the parabola.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Identify the coefficient of the x^2 term and factor it out.
  2. Group the terms that have the same power of x.
  3. Add and subtract the square of half the coefficient of the x term inside the parentheses.
  4. Simplify the expression by combining like terms.

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

A: The vertex form of a quadratic equation is given by y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex of the parabola.

Q: How do I identify the coordinates of the vertex?

A: To identify the coordinates of the vertex, you need to look at the vertex form of the quadratic equation. The value of h is the x-coordinate of the vertex, and the value of k is the y-coordinate of the vertex.

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

A: The vertex of a parabola represents the maximum or minimum value of the function. If the parabola opens upward, the vertex represents the minimum value. If the parabola opens downward, the vertex represents the maximum value.

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

A: Yes, you can use completing the square to solve quadratic equations. By rewriting the equation in vertex form, you can easily identify the x-coordinate of the vertex, which is the solution to the equation.

Q: Are there any other ways to rewrite a quadratic equation in vertex form?

A: Yes, there are other ways to rewrite a quadratic equation in vertex form. One way is to use the formula for the x-coordinate of the vertex, which is x = -b / 2a.

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

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

  • Not factoring out the coefficient of the x^2 term.
  • Not grouping the terms that have the same power of x.
  • Not adding and subtracting the correct constant term inside the parentheses.
  • Not simplifying the expression by combining like terms.

Q: Can I use completing the square to graph quadratic functions?

A: Yes, you can use completing the square to graph quadratic functions. By rewriting the equation in vertex form, you can easily identify the coordinates of the vertex, which can be used to graph the function.

Q: Are there any online resources that can help me learn completing the square?

A: Yes, there are many online resources that can help you learn completing the square, including video tutorials, practice problems, and interactive lessons.

Conclusion

Completing the square is a powerful technique used to rewrite quadratic equations in vertex form. By following the steps outlined in this article, you can easily complete the square and identify the coordinates of the vertex of a parabola. Whether you're a student or a teacher, completing the square is an essential skill that can help you solve quadratic equations and graph quadratic functions.