Converting the Quadratic Function to Vertex Form

Oct 13, 2025 by miodeya.com 11 views

Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on converting a quadratic function from standard form to vertex form and determining the maximum value of the function.

What is Vertex Form?

Vertex form is a way of expressing a quadratic function in the form $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. The vertex form is useful because it allows us to easily identify the maximum or minimum value of the function, as well as the x-coordinate of the vertex.

Converting the Quadratic Function to Vertex Form

The given quadratic function is $y = -x^2 + 2x + 63$ . To convert this function to vertex form, we need to complete the square.

Step 1: Factor out the coefficient of $x^2$

The first step is to factor out the coefficient of $x^2$ , which is -1.

y = -1(x^2 - 2x) + 63

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 $x$ . In this case, half the coefficient of $x$ is -1, and its square is 1.

y = -1(x^2 - 2x + 1 - 1) + 63

Step 3: Factor the perfect square trinomial

Now, we can factor the perfect square trinomial.

y = -1((x - 1)^2 - 1) + 63

Step 4: Simplify the expression

Finally, we can simplify the expression by distributing the -1.

y = -(x - 1)^2 + 1 + 63

Step 5: Write the final vertex form

The final vertex form of the quadratic function is:

y = -(x - 1)^2 + 64

Finding the Maximum Value

The maximum value of the function occurs at the vertex of the parabola. Since the vertex form is $y = a(x - h)^2 + k$ , the maximum value is $k$ . In this case, the maximum value is 64.

Conclusion

In this article, we converted a quadratic function from standard form to vertex form and determined the maximum value of the function. The vertex form is a useful way to express a quadratic function, as it allows us to easily identify the maximum or minimum value of the function, as well as the x-coordinate of the vertex.

Vertex Form: $y = -(x - 1)^2 + 64$ Maximum: $64$

Discussion

What are some other ways to convert a quadratic function from standard form to vertex form?
How can you use the vertex form to determine the maximum or minimum value of a quadratic function?
Can you think of any real-world applications of quadratic functions and their vertex form?
Quadratic Functions: A Q&A Guide =====================================

Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will provide a Q&A guide to help you better understand quadratic functions and their vertex form.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means it has a highest power of two. It can be written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

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

A: The vertex form of a quadratic function is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. The vertex form is useful because it allows us to easily identify the maximum or minimum value of the function, as well as the x-coordinate of the vertex.

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 factoring out the coefficient of $x^2$ , adding and subtracting the square of half the coefficient of $x$ , and then factoring the perfect square trinomial.

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

A: The vertex of a parabola is the point where the parabola changes direction. It is the maximum or minimum point of the function, depending on the sign of the coefficient of $x^2$ . In the vertex form, the vertex is represented by the point $(h, k)$ .

Q: How do I find the maximum or minimum value of a quadratic function?

A: To find the maximum or minimum value of a quadratic function, you need to look at the vertex form. The maximum or minimum value is represented by the constant term $k$ in the vertex form.

Q: Can you provide an example of converting a quadratic function from standard form to vertex form?

A: Let's consider the quadratic function $y = -x^2 + 2x + 63$ . To convert this function to vertex form, we need to complete the square.

Step 1: Factor out the coefficient of $x^2$

y = -1(x^2 - 2x) + 63

Step 2: Add and subtract the square of half the coefficient of $x$

y = -1(x^2 - 2x + 1 - 1) + 63

Step 3: Factor the perfect square trinomial

y = -1((x - 1)^2 - 1) + 63

Step 4: Simplify the expression

y = -(x - 1)^2 + 1 + 63

Step 5: Write the final vertex form

y = -(x - 1)^2 + 64

Q: What are some real-world applications of quadratic functions and their vertex form?

A: Quadratic functions and their vertex form have numerous real-world applications, including:

Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, such as the trajectory of a projectile.
Engineering: Quadratic functions are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic functions are used to model the behavior of economic systems, such as supply and demand curves.

Conclusion

In this article, we provided a Q&A guide to help you better understand quadratic functions and their vertex form. We covered topics such as the definition of a quadratic function, the vertex form, and how to convert a quadratic function from standard form to vertex form. We also discussed the significance of the vertex of a parabola and how to find the maximum or minimum value of a quadratic function.