Rewrite The Quadratic Equation In The Form Y = A ( X − H ) 2 + K Y=a(x-h)^2+k Y = A ( X − H ) 2 + K . Y = 4 X 2 − 16 X + 64 Y=4x^2-16x+64 Y = 4 X 2 − 16 X + 64

Introduction

The quadratic equation is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In its standard form, the quadratic equation is represented as $y=ax^2+bx+c$. However, there are situations where it is more convenient to express the quadratic equation in vertex form, which is given by $y=a(x-h)^2+k$. In this article, we will focus on rewriting the quadratic equation $y=4x^2-16x+64$ in vertex form.

Understanding the Standard Form of a Quadratic Equation

Before we proceed with rewriting the quadratic equation in vertex form, let's briefly review the standard form of a quadratic equation. The standard form of a quadratic equation is given by:

$$y=ax^2+bx+c$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. The graph of a quadratic equation in standard form is a parabola that opens upwards or downwards, depending on the sign of the coefficient $a$.

Rewriting the Quadratic Equation in Vertex Form

To rewrite the quadratic equation $y=4x^2-16x+64$ in vertex form, we need to complete the square. The process of completing the square involves manipulating the quadratic equation to express it in the form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola.

Step 1: Factor Out the Coefficient of $x^2$

The first step in rewriting the quadratic equation in vertex form is to factor out the coefficient of $x^2$, which is $4$. This can be done as follows:

$$y=4(x^2-4x)+64$$

Step 2: Add and Subtract the Square of Half the Coefficient of $x$

Next, we need to add and subtract the square of half the coefficient of $x$ inside the parentheses. Half of the coefficient of $x$ is $-4/2=-2$, and its square is $(-2)^2=4$. Therefore, we can add and subtract $4$ inside the parentheses as follows:

$$y=4(x^2-4x+4-4)+64$$

Step 3: Simplify the Expression

Now, we can simplify the expression by combining like terms:

$$y=4(x^2-4x+4)-16+64$$

$$y=4(x-2)^2+48$$

Step 4: Identify the Vertex

The vertex of the parabola is given by the point $(h,k)$, where $h$ is the value of $x$ that makes the expression inside the parentheses equal to zero, and $k$ is the value of $y$ that results from substituting $x=h$ into the equation. In this case, we can see that $h=2$ and $k=48$. Therefore, the vertex of the parabola is $(2,48)$.

Conclusion

In this article, we have rewritten the quadratic equation $y=4x^2-16x+64$ in vertex form using the process of completing the square. We have shown that the vertex form of the quadratic equation is $y=4(x-2)^2+48$, and we have identified the vertex of the parabola as $(2,48)$. This form of the quadratic equation is useful in various applications, such as graphing the parabola and finding the maximum or minimum value of the function.

Applications of Vertex Form

The vertex form of a quadratic equation has numerous applications in various fields. Some of the applications include:

• Graphing the Parabola: The vertex form of a quadratic equation is useful in graphing the parabola. By plotting the vertex and the axis of symmetry, we can graph the parabola and visualize its shape.
• Finding the Maximum or Minimum Value: The vertex form of a quadratic equation is useful in finding the maximum or minimum value of the function. The vertex of the parabola represents the maximum or minimum value of the function, depending on the sign of the coefficient $a$.
• Solving Systems of Equations: The vertex form of a quadratic equation is useful in solving systems of equations. By substituting the vertex into the equation, we can solve for the values of $x$ and $y$ that satisfy the equation.

Real-World Applications

The vertex form of a quadratic equation has numerous real-world applications. Some of the applications include:

• Physics: The vertex form of a quadratic equation is used in physics to model the motion of objects under the influence of gravity. The vertex of the parabola represents the maximum height of the object.
• Engineering: The vertex form of a quadratic equation is used in engineering to design and optimize systems. The vertex of the parabola represents the optimal value of the system.
• Economics: The vertex form of a quadratic equation is used in economics to model the behavior of economic systems. The vertex of the parabola represents the optimal value of the system.

Conclusion

Introduction

In our previous article, we discussed how to rewrite the quadratic equation $y=4x^2-16x+64$ in vertex form using the process of completing the square. In this article, we will answer some frequently asked questions about quadratic equations in vertex form.

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 $(h,k)$ is the vertex of the parabola.

Q: How do I rewrite a quadratic equation in vertex form?

A: To rewrite a quadratic equation in vertex form, you need to complete the square. This involves manipulating the quadratic equation to express it in the form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola.

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, depending on the sign of the coefficient $a$. It is also the point on the parabola that is closest to the axis of symmetry.

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

A: To find the vertex of a parabola, you need to rewrite the quadratic equation in vertex form. The vertex is given by the point $(h,k)$, where $h$ is the value of $x$ that makes the expression inside the parentheses equal to zero, and $k$ is the value of $y$ that results from substituting $x=h$ into the equation.

Q: What are some real-world applications of quadratic equations in vertex form?

A: Quadratic equations in vertex form have numerous real-world applications, including:

• Physics: The vertex form of a quadratic equation is used in physics to model the motion of objects under the influence of gravity. The vertex of the parabola represents the maximum height of the object.
• Engineering: The vertex form of a quadratic equation is used in engineering to design and optimize systems. The vertex of the parabola represents the optimal value of the system.
• Economics: The vertex form of a quadratic equation is used in economics to model the behavior of economic systems. The vertex of the parabola represents the optimal value of the system.

Q: How do I graph a parabola in vertex form?

A: To graph a parabola in vertex form, you need to plot the vertex and the axis of symmetry. The vertex represents the maximum or minimum value of the function, and the axis of symmetry represents the line that passes through the vertex and is perpendicular to the parabola.

Q: What are some common mistakes to avoid when working with quadratic equations in vertex form?

A: Some common mistakes to avoid when working with quadratic equations in vertex form include:

• Not completing the square: Failing to complete the square can result in an incorrect vertex form of the quadratic equation.
• Not identifying the vertex: Failing to identify the vertex can result in an incorrect graph of the parabola.
• Not using the correct formula: Using the wrong formula can result in an incorrect vertex form of the quadratic equation.

Conclusion

In conclusion, quadratic equations in vertex form are a powerful tool in mathematics and have numerous real-world applications. By understanding how to rewrite a quadratic equation in vertex form and how to find the vertex of a parabola, you can apply these concepts to a variety of problems in physics, engineering, and economics.