Find The Focus, Directrix, Focal Diameter, Vertex, And Axis Of Symmetry For The Parabola Given By The Equation:${ 44(x+4) = (y-7)^2 }$- Focus = □ \square □ - Directrix = □ \square □ - Focal Diameter = □ \square □ - Vertex =

Find the Focus, Directrix, Focal Diameter, Vertex, and Axis of Symmetry for a Parabola

Understanding the Basics of a Parabola

A parabola is a fundamental concept in mathematics, particularly in geometry and algebra. It is a U-shaped curve that can be defined by a quadratic equation. The standard form of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. However, in this article, we will be dealing with a parabola in the form $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

The Given Equation

The given equation of the parabola is $44(x+4) = (y-7)^2$. To find the focus, directrix, focal diameter, vertex, and axis of symmetry, we need to rewrite the equation in the standard form.

Rewriting the Equation

To rewrite the equation, we need to expand the right-hand side and simplify it.

${ 44(x+4) = (y-7)^2 }$ ${ 44x + 176 = y^2 - 14y + 49 }$ ${ y^2 - 14y + 49 = 44x + 176 }$ ${ y^2 - 14y + 49 - 44x - 176 = 0 }$ ${ y^2 - 14y - 44x - 127 = 0 }$

Completing the Square

To complete the square, we need to group the terms with $y$ and $x$ separately.

${ y^2 - 14y - 44x - 127 = 0 }$ ${ (y^2 - 14y) - 44x - 127 = 0 }$ ${ (y^2 - 14y + 49) - 49 - 44x - 127 = 0 }$ ${ (y - 7)^2 - 176 - 44x = 0 }$ ${ (y - 7)^2 = 44x + 176 }$

Rewriting in Standard Form

Now, we can rewrite the equation in the standard form.

${ (y - 7)^2 = 44(x + 4) }$ ${ y - 7 = \pm \sqrt{44(x + 4)} }$ ${ y = 7 \pm \sqrt{44(x + 4)} }$

Finding the Vertex

The vertex of the parabola is given by the point $(h, k)$. In this case, we can see that the vertex is at the point $(-4, 7)$.

Finding the Focus

The focus of the parabola is given by the point $(h, k + \frac{1}{4a})$. In this case, we can see that the focus is at the point $(-4, 7 + \frac{1}{4(44)})$.

Finding the Directrix

The directrix of the parabola is given by the line $y = k - \frac{1}{4a}$. In this case, we can see that the directrix is at the line $y = 7 - \frac{1}{4(44)}$.

Finding the Focal Diameter

The focal diameter of the parabola is given by the distance between the focus and the directrix. In this case, we can see that the focal diameter is given by the distance between the points $(-4, 7 + \frac{1}{4(44)})$ and $(-4, 7 - \frac{1}{4(44)})$.

Finding the Axis of Symmetry

The axis of symmetry of the parabola is given by the line $x = h$. In this case, we can see that the axis of symmetry is at the line $x = -4$.

Conclusion

In this article, we have found the focus, directrix, focal diameter, vertex, and axis of symmetry for the parabola given by the equation $44(x+4) = (y-7)^2$. We have rewritten the equation in the standard form, completed the square, and found the vertex, focus, directrix, focal diameter, and axis of symmetry. The focus is at the point $(-4, 7 + \frac{1}{4(44)})$, the directrix is at the line $y = 7 - \frac{1}{4(44)}$, the focal diameter is given by the distance between the points $(-4, 7 + \frac{1}{4(44)})$ and $(-4, 7 - \frac{1}{4(44)})$, the vertex is at the point $(-4, 7)$, and the axis of symmetry is at the line $x = -4$.

Key Takeaways

• The focus of the parabola is given by the point $(h, k + \frac{1}{4a})$.
• The directrix of the parabola is given by the line $y = k - \frac{1}{4a}$.
• The focal diameter of the parabola is given by the distance between the focus and the directrix.
• The vertex of the parabola is given by the point $(h, k)$.
• The axis of symmetry of the parabola is given by the line $x = h$.

Final Answer

The final answer is:

• Focus = (-4, 7 + 1/176)
• Directrix = y = 7 - 1/176
• Focal diameter = 2 * 1/44
• Vertex = (-4, 7)
• Axis of symmetry = x = -4
  Parabola Focus, Directrix, Focal Diameter, Vertex, and Axis of Symmetry: Q&A

Understanding the Basics of a Parabola

A parabola is a fundamental concept in mathematics, particularly in geometry and algebra. It is a U-shaped curve that can be defined by a quadratic equation. The standard form of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. However, in this article, we will be dealing with a parabola in the form $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q&A Session

Q: What is the focus of a parabola?

A: The focus of a parabola is a point that lies on the axis of symmetry of the parabola. It is given by the point $(h, k + \frac{1}{4a})$.

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

A: To find the focus of a parabola, you need to rewrite the equation of the parabola in the standard form $y = a(x-h)^2 + k$. Then, you can use the formula $(h, k + \frac{1}{4a})$ to find the focus.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. It is given by the line $y = k - \frac{1}{4a}$.

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

A: To find the directrix of a parabola, you need to rewrite the equation of the parabola in the standard form $y = a(x-h)^2 + k$. Then, you can use the formula $y = k - \frac{1}{4a}$ to find the directrix.

Q: What is the focal diameter of a parabola?

A: The focal diameter of a parabola is the distance between the focus and the directrix. It is given by the formula $2 \times \frac{1}{4a}$.

Q: How do I find the focal diameter of a parabola?

A: To find the focal diameter of a parabola, you need to find the focus and the directrix of the parabola. Then, you can use the formula $2 \times \frac{1}{4a}$ to find the focal diameter.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is given by the point $(h, k)$.

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

A: To find the vertex of a parabola, you need to rewrite the equation of the parabola in the standard form $y = a(x-h)^2 + k$. Then, you can use the formula $(h, k)$ to find the vertex.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a line that passes through the vertex of the parabola. It is given by the line $x = h$.

Q: How do I find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, you need to rewrite the equation of the parabola in the standard form $y = a(x-h)^2 + k$. Then, you can use the formula $x = h$ to find the axis of symmetry.

Conclusion

In this article, we have answered some of the most frequently asked questions about the focus, directrix, focal diameter, vertex, and axis of symmetry of a parabola. We have provided step-by-step instructions on how to find these values and have also provided formulas and examples to help you understand the concepts better.

Key Takeaways

• The focus of a parabola is given by the point $(h, k + \frac{1}{4a})$.
• The directrix of a parabola is given by the line $y = k - \frac{1}{4a}$.
• The focal diameter of a parabola is given by the formula $2 \times \frac{1}{4a}$.
• The vertex of a parabola is given by the point $(h, k)$.
• The axis of symmetry of a parabola is given by the line $x = h$.

Final Answer

The final answer is:

• Focus = (-4, 7 + 1/176)
• Directrix = y = 7 - 1/176
• Focal diameter = 2 * 1/44
• Vertex = (-4, 7)
• Axis of symmetry = x = -4