Put The Equation $y = X^2 - 16x + 60$ Into The Form $y = (x - H)^2 + K$.Answer: $ Y = □ Y = \square Y = □ [/tex]

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Introduction

In algebra, quadratic equations are a fundamental concept that can be expressed in various forms. One of the most important forms is the vertex form, which is represented as $y = (x - h)^2 + k$. This form provides valuable information about the graph of the quadratic function, including the vertex, axis of symmetry, and direction of opening. In this article, we will focus on converting the given quadratic equation $y = x^2 - 16x + 60$ into the vertex form.

Understanding the Vertex Form

The vertex form of a quadratic equation is a powerful tool for analyzing and graphing quadratic functions. It is represented as $y = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form provides a clear and concise way to identify the vertex, axis of symmetry, and direction of opening of the parabola.

Completing the Square

To convert the given quadratic equation into the vertex form, we need to complete the square. This process involves manipulating the equation to create a perfect square trinomial. The general steps for completing the square are:

  1. Write the quadratic equation in the form $y = ax^2 + bx + c$.
  2. Move the constant term to the right-hand side of the equation.
  3. Take half of the coefficient of the linear term and square it.
  4. Add the squared value to both sides of the equation.
  5. Factor the left-hand side of the equation to create a perfect square trinomial.

Converting the Given Quadratic Equation

Now, let's apply the steps for completing the square to the given quadratic equation $y = x^2 - 16x + 60$.

Step 1: Move the Constant Term

First, we need to move the constant term to the right-hand side of the equation.

y=x2−16x+60y = x^2 - 16x + 60

0=x2−16x+60−y0 = x^2 - 16x + 60 - y

Step 2: Take Half of the Coefficient of the Linear Term

Next, we need to take half of the coefficient of the linear term and square it.

b=−16b = -16

b2=−162=−8\frac{b}{2} = \frac{-16}{2} = -8

(b2)2=(−8)2=64\left(\frac{b}{2}\right)^2 = (-8)^2 = 64

Step 3: Add the Squared Value to Both Sides

Now, we need to add the squared value to both sides of the equation.

0=x2−16x+60−y+640 = x^2 - 16x + 60 - y + 64

0=x2−16x+64−y+600 = x^2 - 16x + 64 - y + 60

Step 4: Factor the Left-Hand Side

Finally, we need to factor the left-hand side of the equation to create a perfect square trinomial.

0=(x−8)2−y+600 = (x - 8)^2 - y + 60

Step 5: Simplify the Equation

Now, we can simplify the equation by combining like terms.

0=(x−8)2−y+600 = (x - 8)^2 - y + 60

y=(x−8)2−60y = (x - 8)^2 - 60

Final Answer

The final answer is $y = (x - 8)^2 - 60$.

Conclusion

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

A: The vertex form of a quadratic equation is a powerful tool for analyzing and graphing quadratic functions. It is represented as $y = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I convert a quadratic equation to vertex form?

A: To convert a quadratic equation to vertex form, you need to complete the square. This process involves manipulating the equation to create a perfect square trinomial. The general steps for completing the square are:

  1. Write the quadratic equation in the form $y = ax^2 + bx + c$.
  2. Move the constant term to the right-hand side of the equation.
  3. Take half of the coefficient of the linear term and square it.
  4. Add the squared value to both sides of the equation.
  5. Factor the left-hand side of the equation to create a perfect square trinomial.

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

A: The vertex form of a quadratic equation provides valuable information about the graph of the quadratic function, including the vertex, axis of symmetry, and direction of opening. This information is essential for graphing and analyzing quadratic functions.

Q: How do I find the vertex of a quadratic equation in vertex form?

A: To find the vertex of a quadratic equation in vertex form, you need to identify the values of $(h, k)$. The vertex is represented as $(h, k)$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

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

A: The axis of symmetry of a quadratic equation in vertex form is the vertical line that passes through the vertex. The equation of the axis of symmetry is $x = h$, where $h$ is the x-coordinate of the vertex.

Q: How do I determine the direction of opening of a quadratic equation in vertex form?

A: To determine the direction of opening of a quadratic equation in vertex form, you need to examine the sign of the coefficient of the squared term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can I use the vertex form of a quadratic equation to solve systems of equations?

A: Yes, you can use the vertex form of a quadratic equation to solve systems of equations. By substituting the equation of the second quadratic function into the equation of the first quadratic function, you can solve for the x-coordinates of the intersection points.

Q: Are there any limitations to using the vertex form of a quadratic equation?

A: Yes, there are limitations to using the vertex form of a quadratic equation. The vertex form is not suitable for quadratic equations with complex roots or irrational coefficients. In such cases, you may need to use alternative methods, such as the quadratic formula or factoring.

Conclusion

In this article, we have addressed some of the most frequently asked questions about the vertex form of a quadratic equation. We have provided explanations and examples to help you understand the significance and applications of the vertex form. Whether you are a student, teacher, or professional, the vertex form is an essential tool for analyzing and graphing quadratic functions.