The Vertex Form Of The Equation Of A Parabola Is Y = 3 ( X − 4 ) 2 − 22 Y=3(x-4)^2-22 Y = 3 ( X − 4 ) 2 − 22 . What Is The Standard Form Of The Equation?A. Y = 3 X 2 − 18 Y=3x^2-18 Y = 3 X 2 − 18 B. Y = 9 X 2 − 26 Y=9x^2-26 Y = 9 X 2 − 26 C. Y = 3 X 2 − 24 X + 4 Y=3x^2-24x+4 Y = 3 X 2 − 24 X + 4 D. Y = 3 X 2 − 24 X + 26 Y=3x^2-24x+26 Y = 3 X 2 − 24 X + 26

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Understanding the Vertex Form

The vertex form of a parabola is a way to express the equation of a parabola in a specific format. It 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 equation, aa is the coefficient of the squared term, and it determines the direction and width of the parabola. The vertex form is useful for graphing and analyzing parabolas, as it provides a clear and concise way to represent the shape and position of the parabola.

Converting to Standard Form

To convert the vertex form of a parabola to standard form, we need to expand the squared term and simplify the resulting expression. 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. To convert from vertex form to standard form, we can use the following steps:

  1. Expand the squared term: (x−h)2=x2−2hx+h2(x-h)^2 = x^2 - 2hx + h^2
  2. Multiply the expanded term by the coefficient aa: a(x2−2hx+h2)=ax2−2ahx+ah2a(x^2 - 2hx + h^2) = ax^2 - 2ahx + ah^2
  3. Simplify the resulting expression by combining like terms: ax2−2ahx+ah2+kax^2 - 2ahx + ah^2 + k

Applying the Conversion to the Given Equation

Now, let's apply the conversion steps to the given equation: y=3(x−4)2−22y=3(x-4)^2-22. First, we expand the squared term:

(x−4)2=x2−8x+16(x-4)^2 = x^2 - 8x + 16

Next, we multiply the expanded term by the coefficient 33:

3(x2−8x+16)=3x2−24x+483(x^2 - 8x + 16) = 3x^2 - 24x + 48

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

3x2−24x+48−22=3x2−24x+263x^2 - 24x + 48 - 22 = 3x^2 - 24x + 26

Conclusion

In conclusion, the standard form of the equation of a parabola is given by the equation y=ax2+bx+cy=ax^2+bx+c. To convert the vertex form of a parabola to standard form, we need to expand the squared term and simplify the resulting expression. By applying the conversion steps to the given equation, we find that the standard form of the equation is y=3x2−24x+26y=3x^2-24x+26.

Answer

The correct answer is D. y=3x2−24x+26y=3x^2-24x+26.

Additional Examples

Here are a few additional examples of converting the vertex form of a parabola to standard form:

  • y=2(x−2)2+1y=2(x-2)^2+1: y=2x2−8x+5y=2x^2-8x+5
  • y=4(x+3)2−12y=4(x+3)^2-12: y=4x2+24x−36y=4x^2+24x-36
  • y=1(x−1)2+2y=1(x-1)^2+2: y=x2−2x+3y=x^2-2x+3

These examples demonstrate the process of converting the vertex form of a parabola to standard form, and they provide additional practice for converting equations from vertex form to standard form.

Understanding the Vertex Form

The vertex form of a parabola is a way to express the equation of a parabola in a specific format. It 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 equation, aa is the coefficient of the squared term, and it determines the direction and width of the parabola.

Q&A

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 vertex form of a parabola to standard form?

A: To convert the vertex form of a parabola to standard form, you need to expand the squared term and simplify the resulting expression. The steps are:

  1. Expand the squared term: (x−h)2=x2−2hx+h2(x-h)^2 = x^2 - 2hx + h^2
  2. Multiply the expanded term by the coefficient aa: a(x2−2hx+h2)=ax2−2ahx+ah2a(x^2 - 2hx + h^2) = ax^2 - 2ahx + ah^2
  3. Simplify the resulting expression by combining like terms: ax2−2ahx+ah2+kax^2 - 2ahx + ah^2 + k

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

A: To find the vertex of a parabola in vertex form, you need to identify the values of hh and kk in the equation y=a(x−h)2+ky=a(x-h)^2+k. The vertex is given by the point (h,k)(h,k).

Q: What is the significance of the coefficient aa in the vertex form of a parabola?

A: The coefficient aa in the vertex form of a parabola determines the direction and width of the parabola. If aa is positive, the parabola opens upward. If aa is negative, the parabola opens downward.

Q: Can I convert a parabola from standard form to vertex form?

A: Yes, you can convert a parabola from standard form to vertex form by completing the square. The steps are:

  1. Write the equation in standard form: y=ax2+bx+cy=ax^2+bx+c
  2. Complete the square: y=a(x+b2a)2+c−b24ay=a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}
  3. Simplify the resulting expression: y=a(x+b2a)2+ky=a(x+\frac{b}{2a})^2+k

Conclusion

In conclusion, the vertex form of a parabola is a powerful tool for analyzing and graphing parabolas. By understanding the vertex form and how to convert it to standard form, you can gain a deeper understanding of the properties and behavior of parabolas.

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