Which Equation Is $y=(x+3)^2+(x+4)^2$ Rewritten In Vertex Form?A. $y=2\left(x+\frac{7}{2}\right)^2-\frac{1}{4}$B. $y=2\left(x+\frac{7}{2}\right)^2+\frac{1}{2}$C. $y=2(x+7)^2-73$D. $y=(x+7)^2-24$

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Introduction

In mathematics, vertex form is a way of expressing quadratic equations in a specific format that highlights the vertex of the parabola. This form is particularly useful for graphing and analyzing quadratic functions. In this article, we will explore how to rewrite the equation y=(x+3)2+(x+4)2y=(x+3)^2+(x+4)^2 in vertex form and examine the different options provided.

Understanding Vertex Form

Vertex form is a way of expressing a quadratic equation in the format y=a(x−h)2+ky=a(x-h)^2+k, where (h,k)(h,k) is the vertex of the parabola. The vertex form is particularly useful for graphing and analyzing quadratic functions because it allows us to easily identify the vertex and the direction of the parabola.

Rewriting the Equation

To rewrite the equation y=(x+3)2+(x+4)2y=(x+3)^2+(x+4)^2 in vertex form, we need to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial.

Step 1: Expand the Squares

First, we need to expand the squares in the equation:

y=(x+3)2+(x+4)2y=(x+3)^2+(x+4)^2

y=x2+6x+9+x2+8x+16y=x^2+6x+9+x^2+8x+16

Step 2: Combine Like Terms

Next, we need to combine like terms:

y=2x2+14x+25y=2x^2+14x+25

Step 3: Complete the Square

Now, we need to complete the square by adding and subtracting a constant term:

y=2(x2+7x)+25y=2(x^2+7x)+25

y=2(x2+7x+494)+25−492y=2(x^2+7x+\frac{49}{4})+25-\frac{49}{2}

Step 4: Simplify the Expression

Finally, we can simplify the expression by combining like terms:

y=2(x+72)2−14y=2(x+\frac{7}{2})^2-\frac{1}{4}

Comparing Options

Now that we have rewritten the equation in vertex form, we can compare our result with the options provided:

  • A. y=2(x+72)2−14y=2\left(x+\frac{7}{2}\right)^2-\frac{1}{4}
  • B. y=2(x+72)2+12y=2\left(x+\frac{7}{2}\right)^2+\frac{1}{2}
  • C. y=2(x+7)2−73y=2(x+7)^2-73
  • D. y=(x+7)2−24y=(x+7)^2-24

Conclusion

Based on our calculations, we can see that the correct answer is option A: y=2(x+72)2−14y=2\left(x+\frac{7}{2}\right)^2-\frac{1}{4}. This is the only option that matches our rewritten equation in vertex form.

Tips and Tricks

  • When rewriting an equation in vertex form, make sure to complete the square correctly.
  • Use the correct format for vertex form: y=a(x−h)2+ky=a(x-h)^2+k.
  • Pay attention to the signs and coefficients in the equation.

Common Mistakes

  • Failing to complete the square correctly.
  • Using the wrong format for vertex form.
  • Not paying attention to the signs and coefficients in the equation.

Real-World Applications

Vertex form is a useful tool for graphing and analyzing quadratic functions in a variety of real-world applications, including:

  • Physics: to model the motion of objects under the influence of gravity or other forces.
  • Engineering: to design and optimize systems, such as bridges or buildings.
  • Economics: to model the behavior of economic systems and make predictions about future trends.

Conclusion

Introduction

Vertex form is a powerful tool for graphing and analyzing quadratic functions. In our previous article, we explored how to rewrite the equation y=(x+3)2+(x+4)2y=(x+3)^2+(x+4)^2 in vertex form. In this article, we will answer some common questions about vertex form and provide additional tips and tricks for mastering this skill.

Q: What is vertex form?

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

Q: Why is vertex form useful?

A: Vertex form is useful because it allows us to easily identify the vertex and the direction of the parabola. This makes it a powerful tool for graphing and analyzing quadratic functions.

Q: How do I rewrite an equation in vertex form?

A: To rewrite an equation in vertex form, you need to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: What are some common mistakes to avoid when rewriting an equation in vertex form?

A: Some common mistakes to avoid when rewriting an equation in vertex form include:

  • Failing to complete the square correctly.
  • Using the wrong format for vertex form.
  • Not paying attention to the signs and coefficients in the equation.

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

A: To identify the vertex of a parabola in vertex form, look for the values of hh and kk in the equation. The vertex is located at the point (h,k)(h,k).

Q: Can I use vertex form to graph a parabola?

A: Yes, you can use vertex form to graph a parabola. To do this, plot the vertex and then use the direction of the parabola to determine the shape of the graph.

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

A: Vertex form has a variety of real-world applications, including:

  • Physics: to model the motion of objects under the influence of gravity or other forces.
  • Engineering: to design and optimize systems, such as bridges or buildings.
  • Economics: to model the behavior of economic systems and make predictions about future trends.

Q: How do I choose between different options for rewriting an equation in vertex form?

A: When choosing between different options for rewriting an equation in vertex form, look for the option that matches the original equation most closely. This will ensure that you are rewriting the equation correctly.

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

A: Yes, you can use vertex form to solve quadratic equations. To do this, set the equation equal to zero and then solve for the variable.

Conclusion

In conclusion, vertex form is a powerful tool for graphing and analyzing quadratic functions. By mastering this skill, you can solve a variety of problems and apply your knowledge to real-world applications. Remember to pay attention to the signs and coefficients in the equation, and to use the correct format for vertex form.

Additional Resources

  • For more information on vertex form, check out our previous article on rewriting equations in vertex form.
  • For practice problems and exercises, try using online resources such as Khan Academy or Mathway.
  • For real-world applications of vertex form, explore the fields of physics, engineering, and economics.

Common Misconceptions

  • Vertex form is only used for quadratic equations.
  • Vertex form is only used for graphing parabolas.
  • Vertex form is only used for solving quadratic equations.

Conclusion

In conclusion, vertex form is a versatile tool that can be used for a variety of purposes, including graphing and analyzing quadratic functions, solving quadratic equations, and modeling real-world applications. By mastering this skill, you can expand your knowledge and apply your understanding to a wide range of problems.