The Standard Form Of The Equation Of A Parabola Is Y = X 2 − 8 X + 29 Y = X^2 - 8x + 29 Y = X 2 − 8 X + 29 . What Is The Vertex Form Of The Equation?A. Y = ( X − 4 ) 2 + 13 Y = (x - 4)^2 + 13 Y = ( X − 4 ) 2 + 13 B. Y = ( X + 4 ) ( X − 4 ) + 13 Y = (x + 4)(x - 4) + 13 Y = ( X + 4 ) ( X − 4 ) + 13 C. Y = ( X − 4 ) 2 + 28 Y = (x - 4)^2 + 28 Y = ( X − 4 ) 2 + 28 D. Y = ( X − 4 ) 2 + 18 Y = (x - 4)^2 + 18 Y = ( X − 4 ) 2 + 18

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Introduction

In mathematics, a parabola is a quadratic equation that can be represented in various forms. The standard form of a parabola is given by the equation y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants. However, it is often more convenient to represent a parabola in vertex form, which is given by the equation y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In this article, we will explore how to convert the standard form of a parabola to its vertex form.

The Standard Form of a Parabola

The standard form of a parabola is given by the equation y=ax2+bx+cy = ax^2 + bx + c. This form is useful for graphing and analyzing the properties of a parabola. However, it can be difficult to determine the vertex of a parabola from its standard form.

Converting to Vertex Form

To convert the standard form of a parabola to its vertex form, we need to complete the square. This involves rewriting the equation in a way that allows us to easily identify the vertex.

Step 1: Factor Out the Coefficient of x2x^2

The first step in converting the standard form of a parabola to its vertex form is to factor out the coefficient of x2x^2. This will give us a term that we can use to complete the square.

y = x^2 - 8x + 29

Step 2: Complete the Square

Next, we need to complete the square. This involves adding and subtracting a constant term that will allow us to rewrite the equation in a way that reveals the vertex.

y = (x^2 - 8x) + 29

Step 3: Add and Subtract the Constant Term

Now, we need to add and subtract a constant term that will allow us to complete the square.

y = (x^2 - 8x + 16) - 16 + 29

Step 4: Simplify the Equation

Finally, we can simplify the equation by combining like terms.

y = (x - 4)^2 + 13

Conclusion

In this article, we have explored how to convert the standard form of a parabola to its vertex form. We have seen that this involves completing the square, which allows us to rewrite the equation in a way that reveals the vertex. By following these steps, we can easily convert the standard form of a parabola to its vertex form.

Answer

The vertex form of the equation y=x2−8x+29y = x^2 - 8x + 29 is y=(x−4)2+13y = (x - 4)^2 + 13.

Discussion

The vertex form of a parabola is a useful way to represent a quadratic equation. It allows us to easily identify the vertex of the parabola, which is the point at which the parabola changes direction. The vertex form of a parabola is also useful for graphing and analyzing the properties of a parabola.

Example Problems

Problem 1

Convert the standard form of the parabola y=x2+6x+8y = x^2 + 6x + 8 to its vertex form.

Solution

To convert the standard form of the parabola to its vertex form, we need to complete the square.

y = (x^2 + 6x) + 8

Next, we need to add and subtract a constant term that will allow us to complete the square.

y = (x^2 + 6x + 9) - 9 + 8

Finally, we can simplify the equation by combining like terms.

y = (x + 3)^2 - 1

Problem 2

Convert the standard form of the parabola y=x2−4x+3y = x^2 - 4x + 3 to its vertex form.

Solution

To convert the standard form of the parabola to its vertex form, we need to complete the square.

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

Next, we need to add and subtract a constant term that will allow us to complete the square.

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

Finally, we can simplify the equation by combining like terms.

y = (x - 2)^2 - 1

Conclusion

In this article, we have explored how to convert the standard form of a parabola to its vertex form. We have seen that this involves completing the square, which allows us to rewrite the equation in a way that reveals the vertex. By following these steps, we can easily convert the standard form of a parabola to its vertex form.

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Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is given by the equation y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.

Q: How do I convert the standard form of a parabola to its vertex form?

A: To convert the standard form of a parabola to its vertex form, you need to complete the square. This involves rewriting the equation in a way that allows you to easily identify the vertex.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic equation in a way that allows you to easily identify the vertex of the parabola. It involves adding and subtracting a constant term that will allow you to rewrite the equation in a way that reveals the vertex.

Q: How do I complete the square?

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

  1. Factor out the coefficient of x2x^2.
  2. Add and subtract a constant term that will allow you to complete the square.
  3. Simplify the equation by combining like terms.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the parabola changes direction. It is given by the coordinates (h,k)(h, k) in the vertex form of the equation.

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

A: To find the vertex of a parabola, you need to complete the square and rewrite the equation in vertex form. The vertex will be given by the coordinates (h,k)(h, k) in the vertex form of the equation.

Q: What are some common mistakes to avoid when converting the standard form of a parabola to its vertex form?

A: Some common mistakes to avoid when converting the standard form of a parabola to its vertex form include:

  • Not factoring out the coefficient of x2x^2.
  • Not adding and subtracting the correct constant term.
  • Not simplifying the equation by combining like terms.

Q: How do I check my work when converting the standard form of a parabola to its vertex form?

A: To check your work when converting the standard form of a parabola to its vertex form, you can:

  • Plug in a value of xx into the original equation and the vertex form of the equation to see if they are equal.
  • Graph the original equation and the vertex form of the equation to see if they are the same.
  • Use a calculator to graph the original equation and the vertex form of the equation and see if they are the same.

Q: What are some real-world applications of converting the standard form of a parabola to its vertex form?

A: Some real-world applications of converting the standard form of a parabola to its vertex form include:

  • Modeling the trajectory of a projectile.
  • Finding the maximum or minimum value of a quadratic function.
  • Graphing quadratic functions.

Q: How do I use technology to convert the standard form of a parabola to its vertex form?

A: You can use technology such as graphing calculators or computer software to convert the standard form of a parabola to its vertex form. These tools can help you to complete the square and rewrite the equation in vertex form.

Q: What are some tips for mastering the conversion of the standard form of a parabola to its vertex form?

A: Some tips for mastering the conversion of the standard form of a parabola to its vertex form include:

  • Practicing, practicing, practicing.
  • Using technology to help you complete the square and rewrite the equation in vertex form.
  • Checking your work to make sure that you have converted the equation correctly.

Q: How do I know if I have converted the standard form of a parabola to its vertex form correctly?

A: You can check your work by plugging in a value of xx into the original equation and the vertex form of the equation to see if they are equal. You can also graph the original equation and the vertex form of the equation to see if they are the same.