Which Equation Describes A Parabola That Opens Up Or Down And Whose Vertex Is At The Point \[$(h, V)\$\]?A. \[$x = A(y-v)^2 + H\$\]B. \[$x = A(y-h)^2 + V\$\]C. \[$y = A(x-v)^2 + H\$\]D. \[$y = A(x-h)^2 + 4\$\]

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Introduction

In mathematics, a parabola is a type of quadratic curve that can be described by a specific equation. The general 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, when the parabola opens up or down and its vertex is at the point (h,v)(h, v), the equation takes a different form. In this article, we will explore which equation describes a parabola that opens up or down and whose vertex is at the point (h,v)(h, v).

Vertex Form of a Parabola

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. This form is useful when the vertex of the parabola is known, as it allows us to easily identify the equation of the parabola. However, when the parabola opens up or down and its vertex is at the point (h,v)(h, v), the equation takes a different form.

Equation of a Parabola with Vertex at (h, v)

When the parabola opens up or down and its vertex is at the point (h,v)(h, v), the equation of the parabola is given by y=a(x−h)2+vy = a(x-h)^2 + v. This equation is similar to the vertex form of a parabola, but with the vertex at (h,v)(h, v) instead of (h,k)(h, k). The value of aa determines the direction and width of the parabola, while the value of vv determines the vertical position of the vertex.

Comparing the Options

Now that we have identified the correct equation of a parabola with vertex at (h,v)(h, v), let's compare it with the options given:

  • Option A: x=a(y−v)2+hx = a(y-v)^2 + h
  • Option B: x=a(y−h)2+vx = a(y-h)^2 + v
  • Option C: y=a(x−v)2+hy = a(x-v)^2 + h
  • Option D: y=a(x−h)2+4y = a(x-h)^2 + 4

Analysis of Options

  • Option A: This equation is not in the correct form, as it has xx on the left-hand side and yy on the right-hand side. Additionally, the vertex is at (h,v)(h, v), but the equation is not in the correct form to represent this.
  • Option B: This equation is not in the correct form, as it has xx on the left-hand side and yy on the right-hand side. Additionally, the vertex is at (h,v)(h, v), but the equation is not in the correct form to represent this.
  • Option C: This equation is not in the correct form, as it has yy on the left-hand side and xx on the right-hand side. Additionally, the vertex is at (h,v)(h, v), but the equation is not in the correct form to represent this.
  • Option D: This equation is not in the correct form, as it has yy on the left-hand side and xx on the right-hand side. Additionally, the vertex is at (h,v)(h, v), but the equation is not in the correct form to represent this.

Conclusion

In conclusion, the correct equation of a parabola that opens up or down and whose vertex is at the point (h,v)(h, v) is given by y=a(x−h)2+vy = a(x-h)^2 + v. This equation is in the correct form to represent a parabola with vertex at (h,v)(h, v), and it is the only option that matches this description.

Final Answer

Introduction

In our previous article, we discussed the equation of a parabola that opens up or down and whose vertex is at the point (h,v)(h, v). We also compared the options given and identified the correct equation. In this article, we will answer some frequently asked questions related to parabolas and their equations.

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 identify the equation of a parabola with vertex at (h,v)(h, v)?

A: To identify the equation of a parabola with vertex at (h,v)(h, v), you can use the equation y=a(x−h)2+vy = a(x-h)^2 + v. This equation is similar to the vertex form of a parabola, but with the vertex at (h,v)(h, v) instead of (h,k)(h, k).

Q: What is the significance of the value of aa in the equation of a parabola?

A: The value of aa in the equation of a parabola determines the direction and width of the parabola. If aa is positive, the parabola opens up; if aa is negative, the parabola opens down.

Q: How do I determine the vertical position of the vertex of a parabola?

A: To determine the vertical position of the vertex of a parabola, you can use the equation y=a(x−h)2+vy = a(x-h)^2 + v. The value of vv determines the vertical position of the vertex.

Q: Can I use the equation y=a(x−h)2+vy = a(x-h)^2 + v to represent a parabola that opens left or right?

A: No, the equation y=a(x−h)2+vy = a(x-h)^2 + v can only be used to represent a parabola that opens up or down. If you want to represent a parabola that opens left or right, you will need to use a different equation.

Q: How do I graph a parabola with vertex at (h,v)(h, v)?

A: To graph a parabola with vertex at (h,v)(h, v), you can use the equation y=a(x−h)2+vy = a(x-h)^2 + v. You can then plot the vertex at (h,v)(h, v) and use the value of aa to determine the direction and width of the parabola.

Q: Can I use the equation y=a(x−h)2+vy = a(x-h)^2 + v to represent a parabola with a horizontal axis of symmetry?

A: No, the equation y=a(x−h)2+vy = a(x-h)^2 + v can only be used to represent a parabola with a vertical axis of symmetry. If you want to represent a parabola with a horizontal axis of symmetry, you will need to use a different equation.

Conclusion

In conclusion, we have answered some frequently asked questions related to parabolas and their equations. We hope that this article has been helpful in clarifying any confusion you may have had about parabolas and their equations.

Final Answer

The final answer is y=a(x−h)2+vy = a(x-h)^2 + v