Which Equation Is Y = 2 X 2 − 8 X + 9 Y = 2x^2 - 8x + 9 Y = 2 X 2 − 8 X + 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

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 coordinates of the vertex of the parabola. This form is particularly useful for identifying the vertex of a parabola and for graphing quadratic functions.

The Given Equation

The given equation is $y = 2x^2 - 8x + 9$. To rewrite this equation in vertex form, we need to complete the square.

Completing the Square

To complete the square, 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.

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

Now, we can rewrite the equation as:

$$y = 2(x^2 - 4x + 4) - 8 + 9$$

Simplifying the equation, we get:

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

Comparing with the Options

Now that we have rewritten the equation in vertex form, we can compare it with the given options.

• Option A: $y = 2(x - 2)^2 + 9$
• Option B: $y = 2(x - 2)^2 + 5$
• Option C: $y = 2(x - 2)^2 + 1$
• Option D: $y = 2(x - 2)^2 + 17$

Comparing the rewritten equation with the options, we can see that the correct answer is:

Option C: $y = 2(x - 2)^2 + 1$

Conclusion

In this article, we have learned how to rewrite a quadratic equation in vertex form by completing the square. We have also compared the rewritten equation with the given options and identified the correct answer. Vertex form is a useful way of expressing quadratic equations, and it can be particularly helpful for identifying the vertex of a parabola and for graphing quadratic functions.

Key Takeaways

• Vertex form is a way of expressing quadratic equations in the form of $y = a(x - h)^2 + k$.
• To rewrite a quadratic equation in vertex form, we need to complete the square.
• Completing the square involves factoring out the coefficient of the $x^2$ term and adding and subtracting the square of half the coefficient of the $x$ term.
• Vertex form can be particularly helpful for identifying the vertex of a parabola and for graphing quadratic functions.

Frequently Asked Questions

• What is 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 coordinates of the vertex of the parabola.
• How do I rewrite a quadratic equation in vertex form? To rewrite a quadratic equation in vertex form, you need to complete the square by factoring out the coefficient of the $x^2$ term and adding and subtracting the square of half the coefficient of the $x$ term.
• What is the significance of vertex form? Vertex form can be particularly helpful for identifying the vertex of a parabola and for graphing quadratic functions.

Additional Resources

• Khan Academy: Completing the Square
• Mathway: Completing the Square
• Wolfram Alpha: Completing the Square

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Precalculus" by Michael Sullivan
  Vertex Form: A Comprehensive Guide to Quadratic Equations ===========================================================

Q&A: Vertex Form and Quadratic Equations

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 coordinates of the vertex of the parabola.

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 by factoring out the coefficient of the $x^2$ term and adding and subtracting the square of half the coefficient of the $x$ term.

Q: What is the significance of vertex form?

A: Vertex form can be particularly helpful for identifying the vertex of a parabola and for graphing quadratic functions.

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

A: To find the vertex of a parabola in vertex form, you can simply read the coordinates of the vertex from the equation. The vertex is represented by the point $(h, k)$.

Q: Can I use vertex form to graph quadratic functions?

A: Yes, vertex form can be particularly helpful for graphing quadratic functions. By identifying the vertex of the parabola, you can determine the direction and shape of the graph.

Q: How do I determine the direction of the parabola in vertex form?

A: To determine the direction of the parabola in vertex form, you can look at the sign of the coefficient $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: Can I use vertex form to solve quadratic equations?

A: Yes, vertex form can be used to solve quadratic equations. By setting the equation equal to zero and solving for $x$, you can find the solutions to the equation.

Q: How do I solve quadratic equations in vertex form?

A: To solve quadratic equations in vertex form, you can set the equation equal to zero and solve for $x$. This will give you the solutions to the equation.

Q: What are some common mistakes to avoid when working with vertex form?

A: Some common mistakes to avoid when working with 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
• Not reading the coordinates of the vertex correctly
• Not determining the direction of the parabola correctly

Q: How can I practice working with vertex form?

A: You can practice working with vertex form by:

• Completing the square with different quadratic equations
• Graphing quadratic functions in vertex form
• Solving quadratic equations in vertex form
• Identifying the vertex of a parabola in vertex form

Q: What are some real-world applications of vertex form?

A: Some real-world applications of vertex form include:

• Modeling population growth and decline
• Analyzing the motion of objects under the influence of gravity
• Determining the maximum or minimum value of a function
• Graphing quadratic functions in different contexts

Conclusion

In this article, we have explored the concept of vertex form and its applications in quadratic equations. We have also answered some common questions about vertex form and provided some tips for practicing and applying vertex form in different contexts. Whether you are a student or a professional, vertex form is an essential tool for working with quadratic equations and functions.