Rewriting the Quadratic Function in Vertex Form

Oct 13, 2025 by windailys.com 71 views

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to rewrite them in vertex form is crucial for solving various mathematical problems. In this article, we will focus on completing the square to rewrite the quadratic function in vertex form. We will use the given quadratic function $y = x^2 + x + 8$ as an example and guide you through the step-by-step process.

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 for identifying the maximum or minimum value of the quadratic function, as well as the direction of the parabola's opening.

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 complete the square, 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 $y = x^2 + x + 8 - 8$ .

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, the coefficient of $x$ is 1, so we need to add and subtract $(1/2)^2 = 1/4$ . This will give us $y = x^2 + x + 8 - 8 + 1/4 - 1/4$ .

Step 3: Factor the Perfect Square Trinomial

Now, we can factor the perfect square trinomial on the left-hand side of the equation. This will give us $y = (x + 1/2)^2 + 8 - 1/4$ .

Step 4: Simplify the Expression

Finally, we can simplify the expression by evaluating the constant term. This will give us $y = (x + 1/2)^2 + 31/4$ .

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 quadratic function is $y = (x + 1/2)^2 + 31/4$ . This tells us that the vertex of the parabola is at the point $(-1/2, 31/4)$ .

Example 2: Rewriting a Quadratic Function in Vertex Form

Let's consider another example of a quadratic function: $y = x^2 - 4x + 3$ . We can complete the square to rewrite this function in vertex form.

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

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, the coefficient of $x$ is -4, so we need to add and subtract $(-4/2)^2 = 4$ . This will give us $y = x^2 - 4x + 3 - 3 + 4 - 4$ .

Step 3: Factor the Perfect Square Trinomial

Now, we can factor the perfect square trinomial on the left-hand side of the equation. This will give us $y = (x - 2)^2 + 3 - 4$ .

Step 4: Simplify the Expression

Finally, we can simplify the expression by evaluating the constant term. This will give us $y = (x - 2)^2 - 1$ .

Conclusion

In this article, we have learned how to complete the square to rewrite a quadratic function in vertex form. We have used two examples to illustrate the process, and we have seen how to identify the vertex of the parabola from the vertex form of the quadratic function. Completing the square is a powerful technique that can be used to solve a wide range of mathematical problems, and we hope that this article has provided you with a clear understanding of how to use it.

Frequently Asked Questions

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 complete the square to rewrite a quadratic function in vertex form?

A: To complete the square, you need to follow these steps:

Move the constant term to the right-hand side of the equation.
Add and subtract the square of half the coefficient of $x$ .
Factor the perfect square trinomial on the left-hand side of the equation.
Simplify the expression by evaluating the constant term.

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

A: The vertex of the parabola is the point $(h, k)$ in the vertex form of the quadratic function.

Glossary

Vertex form: 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.
Completing the square: A technique used to rewrite a quadratic function in vertex form.
Perfect square trinomial: A trinomial that can be factored into the square of a binomial.
Vertex: The point $(h, k)$ in the vertex form of a quadratic function, where $(h, k)$ is the vertex of the parabola.
Quadratic Functions: A Q&A Guide =====================================

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide to help you better understand quadratic functions and how to work with them.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that the highest power of the variable (usually x) is two. Quadratic functions are often 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. This form is useful for identifying the maximum or minimum value of the quadratic function, as well as the direction of the parabola's opening.

Q: How do I complete the square to rewrite a quadratic function in vertex form?

A: To complete the square, you need to follow these steps:

Move the constant term to the right-hand side of the equation.
Add and subtract the square of half the coefficient of $x$ .
Factor the perfect square trinomial on the left-hand side of the equation.
Simplify the expression by evaluating the constant term.

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

A: The vertex of the parabola is the point $(h, k)$ in the vertex form of the quadratic function. This point represents the maximum or minimum value of the quadratic function, depending on the direction of the parabola's opening.

Q: How do I determine the direction of the parabola's opening in the vertex form of a quadratic function?

A: To determine the direction of the parabola's opening, you need to look at the coefficient of the squared term ( $a$ ). If $a$ is positive, the parabola opens upward, and if $a$ is negative, the parabola opens downward.

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

A: The axis of symmetry is the vertical line that passes through the vertex of the parabola. It is represented by the equation $x = h$ , where $(h, k)$ is the vertex of the parabola.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for $x$ . This will give you the x-coordinates of the points where the graph of the function intersects the x-axis.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.

Q: Can a quadratic function have more than one x-intercept?

A: Yes, a quadratic function can have more than one x-intercept. This occurs when the graph of the function intersects the x-axis at two or more points.

Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts. This occurs when the graph of the function does not intersect the x-axis at any point.

Conclusion

In this article, we have provided a Q&A guide to help you better understand quadratic functions and how to work with them. We have covered topics such as the vertex form of a quadratic function, completing the square, and finding the x-intercepts of a quadratic function. We hope that this guide has been helpful in your understanding of quadratic functions.

Frequently Asked Questions

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 complete the square to rewrite a quadratic function in vertex form?

A: To complete the square, you need to follow these steps:

Move the constant term to the right-hand side of the equation.
Add and subtract the square of half the coefficient of $x$ .
Factor the perfect square trinomial on the left-hand side of the equation.
Simplify the expression by evaluating the constant term.

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

A: The vertex of the parabola is the point $(h, k)$ in the vertex form of the quadratic function.

Glossary

Quadratic function: A polynomial function of degree two, which means that the highest power of the variable (usually x) is two.
Vertex form: 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.
Completing the square: A technique used to rewrite a quadratic function in vertex form.
Perfect square trinomial: A trinomial that can be factored into the square of a binomial.
Vertex: The point $(h, k)$ in the vertex form of a quadratic function, where $(h, k)$ is the vertex of the parabola.