Which Equation Is $y=2x^2-8x+9$ Rewritten In Vertex Form?A. Y = 2 ( X − 2 ) 2 + 9 Y=2(x-2)^2+9 Y = 2 ( X − 2 ) 2 + 9 B. Y = 2 ( X − 2 ) 2 + 5 Y=2(x-2)^2+5 Y = 2 ( X − 2 ) 2 + 5 C. Y = 2 ( X − 2 ) 2 + 1 Y=2(x-2)^2+1 Y = 2 ( X − 2 ) 2 + 1 D. Y = 2 ( X − 2 ) 2 + 17 Y=2(x-2)^2+17 Y = 2 ( X − 2 ) 2 + 17

Mar 12, 2025 by ADMIN 302 views

**Rewriting Quadratic Equations in Vertex Form**

Understanding Vertex Form

Vertex form is a way of expressing quadratic equations in the form of $y=a(x-h)^2+k$ , where $(h,k)$ represents the vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola's opening. In this article, we will explore how to rewrite a given quadratic equation in vertex form.

The Given Equation

The given quadratic equation is $y=2x^2-8x+9$ . Our goal is to rewrite this equation in vertex form.

Completing the Square

To rewrite the equation in vertex form, we need to complete the square. This involves manipulating the equation to create a perfect square trinomial. We start by factoring out the coefficient of the $x^2$ term, which is 2.

y = 2(x^2 - 4x) + 9

Next, we add and subtract the square of half the coefficient of the $x$ term inside the parentheses. The coefficient of the $x$ term is -4, so half of it is -2, and its square is 4.

y = 2(x^2 - 4x + 4 - 4) + 9

Now, we can rewrite the equation as:

y = 2((x - 2)^2 - 4) + 9

Simplifying the Equation

We can simplify the equation by distributing the 2 to the terms inside the parentheses.

y = 2(x - 2)^2 - 8 + 9

Combining the constant terms, we get:

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

Conclusion

We have successfully rewritten the given quadratic equation $y=2x^2-8x+9$ in vertex form as $y=2(x-2)^2+1$ . This form allows us to easily identify the vertex of the parabola as $(2,1)$ and determine the direction of the parabola's opening.

Answer

The correct answer is C. $y=2(x-2)^2+1$ .

Why is this important?

Understanding how to rewrite quadratic equations in vertex form is crucial in various mathematical applications, such as graphing parabolas, finding the maximum or minimum value of a quadratic function, and solving systems of equations. By mastering this concept, you will be able to tackle more complex mathematical problems with confidence.

Real-World Applications

Vertex form has numerous real-world applications, including:

Physics and Engineering: Vertex form is used to model the motion of objects under the influence of gravity, springs, and other forces.
Economics: Vertex form is used to model the relationship between variables, such as supply and demand, and to determine the optimal price or quantity.
Computer Science: Vertex form is used in algorithms for solving linear and quadratic equations, which is essential in computer graphics, game development, and other fields.

Tips and Tricks

Practice, Practice, Practice: The more you practice rewriting quadratic equations in vertex form, the more comfortable you will become with the process.
Use Online Resources: There are many online resources available that can help you practice and learn how to rewrite quadratic equations in vertex form.
Watch Video Tutorials: Video tutorials can provide a visual explanation of the process and help you understand the concept better.

Conclusion

Q: What is vertex form?

A: Vertex form is a way of expressing quadratic equations in the form of $y=a(x-h)^2+k$ , where $(h,k)$ represents the vertex of the parabola.

Q: Why is vertex form important?

A: Vertex form is important because it allows us to easily identify the vertex of the parabola and determine the direction of the parabola's opening. This is crucial in various mathematical applications, such as graphing parabolas, finding the maximum or minimum value of a quadratic function, and solving systems of equations.

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 equation to create a perfect square trinomial. You start by factoring out the coefficient of the $x^2$ term, then add and subtract the square of half the coefficient of the $x$ term inside the parentheses.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is represented by the coordinates $(h,k)$ in vertex form.

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 then represented by the coordinates $(h,k)$ .

Q: What is the difference between standard form and vertex form?

A: Standard form is the original form of a quadratic equation, while vertex form is a rewritten form that highlights the vertex of the parabola.

Q: Can I use vertex form to solve systems of equations?

A: Yes, vertex form can be used to solve systems of equations. By rewriting the quadratic equations in vertex form, you can easily identify the vertices of the parabolas and determine the points of intersection.

Q: Are there any online resources available to help me learn vertex form?

A: Yes, there are many online resources available that can help you learn vertex form, including video tutorials, practice problems, and interactive quizzes.

Q: Can I use vertex form to model real-world applications?

A: Yes, vertex form can be used to model real-world applications, such as the motion of objects under the influence of gravity, the relationship between variables in economics, and the behavior of algorithms in computer science.

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

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

Not factoring out the coefficient of the $x^2$ term
Not adding and subtracting the square of half the coefficient of the $x$ term inside the parentheses
Not simplifying the equation after completing the square

Q: How can I practice rewriting quadratic equations in vertex form?

A: You can practice rewriting quadratic equations in vertex form by using online resources, such as video tutorials and practice problems, or by working with a tutor or teacher. You can also try rewriting quadratic equations on your own and checking your work with a calculator or online tool.

Q: What are some advanced topics related to vertex form?

A: Some advanced topics related to vertex form include:

Quadratic equations with complex coefficients: These are quadratic equations that have complex coefficients, which can be rewritten in vertex form using complex numbers.
Quadratic equations with rational coefficients: These are quadratic equations that have rational coefficients, which can be rewritten in vertex form using rational numbers.
Quadratic equations with irrational coefficients: These are quadratic equations that have irrational coefficients, which can be rewritten in vertex form using irrational numbers.

Conclusion

Vertex form is a powerful tool that can help you solve a wide range of mathematical problems. By mastering this concept, you will be able to tackle more complex mathematical problems with confidence and apply your knowledge to real-world applications.