Complete The Square To Rewrite The Quadratic Function In Vertex Form:${ Y = X^2 - 7x + 1 }$

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to rewrite them in vertex form is crucial for various applications in algebra, calculus, and other fields. In this article, we will explore the process of completing the square to rewrite a quadratic function in vertex form. We will use the given quadratic function $y = x^2 - 7x + 1$ as an example to illustrate the steps involved.

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. This form is useful because it allows us to easily identify the vertex of the parabola, which is the maximum or minimum point of the function.

Completing the Square

Completing the square is a technique used to rewrite a quadratic function in vertex form. The process involves creating a perfect square trinomial from the given quadratic function. To do this, we need to follow these steps:

Step 1: Move the Constant Term

The first step is to move the constant term to the right-hand side of the equation. This will give us $y = x^2 - 7x + 1 - 1$.

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 $x$ to the equation. The coefficient of $x$ is $-7$, so half of this is $-\frac{7}{2}$. We add and subtract the square of this value, which is $\left(-\frac{7}{2}\right)^2 = \frac{49}{4}$.

y = x^2 - 7x + 1 - 1
y = x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 1

Step 3: Factor the Perfect Square Trinomial

Now, we can factor the perfect square trinomial on the left-hand side of the equation. This gives us $y = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 1$.

Step 4: Simplify the Equation

Finally, we can simplify the equation by combining the constant terms on the right-hand side. This gives us $y = \left(x - \frac{7}{2}\right)^2 - \frac{45}{4}$.

Rewriting the Quadratic Function in Vertex Form

Now that we have completed the square, we can rewrite the quadratic function in vertex form. The vertex form of the function is $y = \left(x - \frac{7}{2}\right)^2 - \frac{45}{4}$.

Identifying the Vertex

The vertex of the parabola is the point $(h, k)$ in the vertex form of the function. In this case, the vertex is $\left(\frac{7}{2}, -\frac{45}{4}\right)$.

Conclusion

In this article, we have explored the process of completing the square to rewrite a quadratic function in vertex form. We used the given quadratic function $y = x^2 - 7x + 1$ as an example to illustrate the steps involved. By following these steps, we were able to rewrite the function in vertex form and identify the vertex of the parabola.

Example Problems

Problem 1

Rewrite the quadratic function $y = x^2 + 5x - 3$ in vertex form using the method of completing the square.

Solution

To rewrite the function in vertex form, we need to complete the square. We start by moving the constant term to the right-hand side of the equation:

y = x^2 + 5x - 3
y = x^2 + 5x + \frac{25}{4} - \frac{25}{4} - 3

Next, we add and subtract the square of half the coefficient of $x$:

y = x^2 + 5x + \frac{25}{4} - \frac{25}{4} - 3
y = \left(x + \frac{5}{2}\right)^2 - \frac{25}{4} - 3

Finally, we simplify the equation by combining the constant terms on the right-hand side:

y = \left(x + \frac{5}{2}\right)^2 - \frac{25}{4} - 3
y = \left(x + \frac{5}{2}\right)^2 - \frac{37}{4}

The vertex form of the function is $y = \left(x + \frac{5}{2}\right)^2 - \frac{37}{4}$.

Problem 2

Rewrite the quadratic function $y = x^2 - 2x + 4$ in vertex form using the method of completing the square.

Solution

To rewrite the function in vertex form, we need to complete the square. We start by moving the constant term to the right-hand side of the equation:

y = x^2 - 2x + 4
y = x^2 - 2x + 1 - 1 + 4

Next, we add and subtract the square of half the coefficient of $x$:

y = x^2 - 2x + 1 - 1 + 4
y = \left(x - 1\right)^2 - 1 + 4

Finally, we simplify the equation by combining the constant terms on the right-hand side:

y = \left(x - 1\right)^2 - 1 + 4
y = \left(x - 1\right)^2 + 3

The vertex form of the function is $y = \left(x - 1\right)^2 + 3$.

Final Thoughts

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Introduction

In our previous article, we explored the process of completing the square to rewrite a quadratic function in vertex form. In this article, we will answer some frequently asked questions about quadratic functions in vertex form.

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.

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

A: To rewrite a quadratic function in vertex form, you need to complete the square. This involves moving the constant term to the right-hand side of the equation, adding and subtracting the square of half the coefficient of $x$, and then simplifying the equation.

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

A: The vertex of a parabola is the maximum or minimum point of the function. In the vertex form of a quadratic function, the vertex is represented by the point $(h, k)$.

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

A: To find the vertex of a parabola, you need to rewrite the quadratic function in vertex form. The vertex is then represented by the point $(h, k)$ in the vertex form of the function.

Q: Can I use the method of completing the square to rewrite any quadratic function in vertex form?

A: Yes, you can use the method of completing the square to rewrite any quadratic function in vertex form. However, you need to make sure that the quadratic function is in the form $y = ax^2 + bx + c$.

Q: What are some common mistakes to avoid when rewriting a quadratic function in vertex form?

A: Some common mistakes to avoid when rewriting a quadratic function in vertex form include:

• Not moving the constant term to the right-hand side of the equation
• Not adding and subtracting the square of half the coefficient of $x$
• Not simplifying the equation correctly

Q: Can I use a calculator to rewrite a quadratic function in vertex form?

A: Yes, you can use a calculator to rewrite a quadratic function in vertex form. However, it's always a good idea to check your work by hand to make sure that you understand the process.

Q: How do I apply the concept of vertex form to real-world problems?

A: The concept of vertex form can be applied to a wide range of real-world problems, including:

• Modeling population growth
• Analyzing the motion of objects
• Optimizing functions

Example Problems

Problem 1

Rewrite the quadratic function $y = x^2 + 4x - 3$ in vertex form using the method of completing the square.

Solution

To rewrite the function in vertex form, we need to complete the square. We start by moving the constant term to the right-hand side of the equation:

y = x^2 + 4x - 3
y = x^2 + 4x + 4 - 4 - 3

Next, we add and subtract the square of half the coefficient of $x$:

y = x^2 + 4x + 4 - 4 - 3
y = \left(x + 2\right)^2 - 4 - 3

Finally, we simplify the equation by combining the constant terms on the right-hand side:

y = \left(x + 2\right)^2 - 4 - 3
y = \left(x + 2\right)^2 - 7

The vertex form of the function is $y = \left(x + 2\right)^2 - 7$.

Problem 2

Rewrite the quadratic function $y = x^2 - 6x + 2$ in vertex form using the method of completing the square.

Solution

To rewrite the function in vertex form, we need to complete the square. We start by moving the constant term to the right-hand side of the equation:

y = x^2 - 6x + 2
y = x^2 - 6x + 9 - 9 + 2

Next, we add and subtract the square of half the coefficient of $x$:

y = x^2 - 6x + 9 - 9 + 2
y = \left(x - 3\right)^2 - 9 + 2

Finally, we simplify the equation by combining the constant terms on the right-hand side:

y = \left(x - 3\right)^2 - 9 + 2
y = \left(x - 3\right)^2 - 7

The vertex form of the function is $y = \left(x - 3\right)^2 - 7$.

Final Thoughts

In this article, we have answered some frequently asked questions about quadratic functions in vertex form. We have also provided example problems to help illustrate the process of rewriting a quadratic function in vertex form. By following these steps, you can apply the concept of vertex form to a wide range of real-world problems.