If F ( X ) = ( X − 2 ) 2 + 1 F(x)=(x-2)^2+1 F ( X ) = ( X − 2 ) 2 + 1 , Which Point Is The Vertex Of The Parabola?A. ( 0 , 1 (0,1 ( 0 , 1 ] B. ( 2 , 1 (2,1 ( 2 , 1 ] C. ( 0 , 2 (0,2 ( 0 , 2 ] D. ( 2 , 0 (2,0 ( 2 , 0 ]

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Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in various forms, including the standard form, vertex form, and factored form. The vertex form of a parabola is given by the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) represents the coordinates of the vertex. In this article, we will explore how to find the vertex of a parabola using the given equation f(x)=(x−2)2+1f(x) = (x - 2)^2 + 1.

What is a Parabola?

A parabola is a U-shaped curve that can be opened upwards or downwards. It is a quadratic equation that can be represented in various forms, including the standard form, vertex form, and factored form. The vertex form of a parabola is given by the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) represents the coordinates of the vertex.

The Vertex Form of a Parabola

The vertex form of a parabola is given by the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k. In this equation, (h,k)(h, k) represents the coordinates of the vertex. The value of hh represents the x-coordinate of the vertex, while the value of kk represents the y-coordinate of the vertex.

Finding the Vertex of a Parabola

To find the vertex of a parabola, we need to identify the values of hh and kk in the vertex form of the equation. In the given equation f(x)=(x−2)2+1f(x) = (x - 2)^2 + 1, we can see that the value of hh is 2 and the value of kk is 1.

The Vertex of the Parabola

Using the values of hh and kk, we can determine the coordinates of the vertex. The x-coordinate of the vertex is given by the value of hh, which is 2. The y-coordinate of the vertex is given by the value of kk, which is 1. Therefore, the coordinates of the vertex are (2,1)(2, 1).

Conclusion

In conclusion, the vertex of the parabola given by the equation f(x)=(x−2)2+1f(x) = (x - 2)^2 + 1 is (2,1)(2, 1). This is because the value of hh is 2 and the value of kk is 1. Therefore, the correct answer is B. (2,1)(2, 1).

Frequently Asked Questions

Q: What is the vertex form of a parabola?

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

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

A: To find the vertex of a parabola, you need to identify the values of hh and kk in the vertex form of the equation.

Q: What are the coordinates of the vertex of the parabola given by the equation f(x)=(x−2)2+1f(x) = (x - 2)^2 + 1?

A: The coordinates of the vertex of the parabola given by the equation f(x)=(x−2)2+1f(x) = (x - 2)^2 + 1 are (2,1)(2, 1).

References

Glossary

  • Vertex: The point on a parabola that is the lowest or highest point on the curve.
  • Parabola: A U-shaped curve that can be opened upwards or downwards.
  • Vertex Form: A form of a parabola that is given by the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) represents the coordinates of the vertex.

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Introduction

In our previous article, we discussed how to find the vertex of a parabola using the given equation f(x)=(x−2)2+1f(x) = (x - 2)^2 + 1. In this article, we will answer some of the most frequently asked questions about the vertex of a parabola.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the curve that is the lowest or highest point on the parabola. It is represented by the coordinates (h,k)(h, k) in the vertex form of the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k.

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

A: To find the vertex of a parabola, you need to identify the values of hh and kk in the vertex form of the equation. The value of hh represents the x-coordinate of the vertex, while the value of kk represents the y-coordinate of the vertex.

Q: What is the vertex form of a parabola?

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

Q: How do I determine the coordinates of the vertex?

A: To determine the coordinates of the vertex, you need to identify the values of hh and kk in the vertex form of the equation. The x-coordinate of the vertex is given by the value of hh, while the y-coordinate of the vertex is given by the value of kk.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is significant because it represents the lowest or highest point on the curve. It is also used to determine the direction of the parabola, whether it opens upwards or downwards.

Q: Can the vertex of a parabola be negative?

A: Yes, the vertex of a parabola can be negative. The y-coordinate of the vertex, represented by the value of kk, can be negative or positive, depending on the equation of the parabola.

Q: How do I graph a parabola with a negative vertex?

A: To graph a parabola with a negative vertex, you need to use the vertex form of the equation and plot the vertex on the coordinate plane. The parabola will open downwards if the value of aa is negative.

Q: Can the vertex of a parabola be a point of inflection?

A: Yes, the vertex of a parabola can be a point of inflection. A point of inflection is a point on the curve where the concavity changes, and the vertex can be one of these points.

Q: How do I determine if the vertex of a parabola is a point of inflection?

A: To determine if the vertex of a parabola is a point of inflection, you need to examine the second derivative of the equation. If the second derivative is zero at the vertex, then the vertex is a point of inflection.

Conclusion

In conclusion, the vertex of a parabola is an important concept in mathematics that represents the lowest or highest point on the curve. It is used to determine the direction of the parabola and can be a point of inflection. We hope that this article has answered some of the most frequently asked questions about the vertex of a parabola.

Frequently Asked Questions

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the curve that is the lowest or highest point on the parabola.

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

A: To find the vertex of a parabola, you need to identify the values of hh and kk in the vertex form of the equation.

Q: What is the vertex form of a parabola?

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

References

Glossary

  • Vertex: The point on a parabola that is the lowest or highest point on the curve.
  • Parabola: A U-shaped curve that can be opened upwards or downwards.
  • Vertex Form: A form of a parabola that is given by the equation f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) represents the coordinates of the vertex.