Determining Whether the Vertex is a Minimum or a Maximum

Oct 13, 2025 by windailys.com 0 views

Introduction

In mathematics, the completing-the-square method is a powerful technique used to find the vertex of a quadratic function. This method involves rewriting the quadratic function in a specific form that allows us to easily identify the vertex. In this article, we will use the completing-the-square method to find the vertex of the function $f(x) = -3x^2 + 6x - 2$ and determine whether it is a minimum or a maximum.

What is the Completing-the-Square Method?

The completing-the-square method is a technique used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. This form is called the vertex form of a quadratic function. The completing-the-square method involves manipulating the quadratic function to create a perfect square trinomial, which can be written in the form $(x - h)^2$ .

Step 1: Write the Quadratic Function in General Form

The given quadratic function is $f(x) = -3x^2 + 6x - 2$ . This is the general form of a quadratic function, which is $f(x) = ax^2 + bx + c$ . In this case, $a = -3$ , $b = 6$ , and $c = -2$ .

Step 2: Divide the Quadratic Function by the Coefficient of $x^2$

To complete the square, we need to divide the quadratic function by the coefficient of $x^2$ , which is $-3$ . This will give us a new function with a leading coefficient of $1$ . The new function is $f(x) = -x^2 + 2x - \frac{2}{3}$ .

Step 3: Add and Subtract the Square of Half the Coefficient of $x$

To complete the square, we need to add and subtract the square of half the coefficient of $x$ . In this case, the coefficient of $x$ is $2$ , so half of it is $1$ . The square of $1$ is $1$ , so we add and subtract $1$ from the function. The new function is $f(x) = -x^2 + 2x - 1 + \frac{1}{3}$ .

Step 4: Write the Function in Vertex Form

Now that we have added and subtracted the square of half the coefficient of $x$ , we can write the function in vertex form. The function is $f(x) = -\left(x - 1\right)^2 + \frac{1}{3}$ .

Finding the Vertex

The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. In this case, the vertex form of the function is $f(x) = -\left(x - 1\right)^2 + \frac{1}{3}$ . Therefore, the vertex of the parabola is $(1, \frac{1}{3})$ .

Determining Whether the Vertex is a Minimum or a Maximum

The coefficient of the squared term in the vertex form of the function is $-1$ . Since the coefficient is negative, the vertex is a maximum.

Conclusion

In this article, we used the completing-the-square method to find the vertex of the function $f(x) = -3x^2 + 6x - 2$ . We determined that the vertex is a maximum and is located at $(1, \frac{1}{3})$ . This is the correct answer, so the correct option is A. Maximum at (1, 1) is incorrect.

Final Answer

Introduction

In our previous article, we used the completing-the-square method to find the vertex of the function $f(x) = -3x^2 + 6x - 2$ . In this article, we will answer some frequently asked questions about the completing-the-square method.

Q: What is the completing-the-square method?

A: The completing-the-square method is a technique used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. This form is called the vertex form of a quadratic function.

Q: How do I know if the vertex is a minimum or a maximum?

A: To determine whether the vertex is a minimum or a maximum, you need to look at the coefficient of the squared term in the vertex form of the function. If the coefficient is positive, the vertex is a minimum. If the coefficient is negative, the vertex is a maximum.

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

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

Q: How do I find the vertex of a quadratic function using the completing-the-square method?

A: To find the vertex of a quadratic function using the completing-the-square method, you need to follow these steps:

Write the quadratic function in general form.
Divide the quadratic function by the coefficient of $x^2$ .
Add and subtract the square of half the coefficient of $x$ .
Write the function in vertex form.

Q: What is the difference between the completing-the-square method and the factoring method?

A: The completing-the-square method and the factoring method are two different techniques used to find the vertex of a quadratic function. The factoring method involves factoring the quadratic function into the product of two binomials, while the completing-the-square method involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$ .

Q: When should I use the completing-the-square method?

A: You should use the completing-the-square method when you need to find the vertex of a quadratic function and the function is not easily factored.

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

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

Not dividing the quadratic function by the coefficient of $x^2$ .
Not adding and subtracting the square of half the coefficient of $x$ .
Not writing the function in vertex form.

Conclusion

In this article, we answered some frequently asked questions about the completing-the-square method. We hope that this article has been helpful in understanding the completing-the-square method and how to use it to find the vertex of a quadratic function.

Final Tips

Make sure to follow the steps of the completing-the-square method carefully.
Check your work by plugging the vertex back into the original function.
Practice using the completing-the-square method to find the vertex of different quadratic functions.

Common Quadratic Functions

Here are some common quadratic functions that you can use to practice the completing-the-square method:

$f(x) = x^2 + 4x + 4$
$f(x) = -2x^2 + 6x - 3$
$f(x) = x^2 - 2x - 3$

Practice Problems

Here are some practice problems that you can use to practice the completing-the-square method:

Find the vertex of the function $f(x) = x^2 + 6x + 9$ .
Find the vertex of the function $f(x) = -3x^2 + 12x - 15$ .
Find the vertex of the function $f(x) = x^2 - 4x - 5$ .

We hope that this article has been helpful in understanding the completing-the-square method and how to use it to find the vertex of a quadratic function.