Understanding Parabolas in Algebra

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Questions and Answers

What is the eccentricity of a parabola?

1 (correct)

0.5

Which equation represents a parabola with its principal axis parallel to the x-axis?

$y - k = -4a(x - h)$

$y - k = 4a(x - h)$ (correct)

$y - k^2 = 4a(x - h)$

$x - h = 4a(y - k)$

What defines the focus of a parabola?

It is the average distance of all points from the directrix.

It is located halfway between the directrix and the vertex.

It is the point from which the parabola opens. (correct)

It is always placed at the origin of the graph.

What is the standard equation of a parabola that opens upward?

$x - h^2 = 4a(y - k)$ (B) Signup and view all the answers

What is the length of the latus rectum of a parabola?

$4a$ (D) Signup and view all the answers

Which of the following accurately describes the vertex of a parabola?

It is equidistant from the focus and the directrix. (A) Signup and view all the answers

Which parabola opens to the left based on its standard equation?

$y - k^2 = -4a(x - h)$ (C) Signup and view all the answers

Which components are essential parts of a parabola?

Focus, Directrix, Vertex, Latus Rectum (C) Signup and view all the answers

What is the principal axis of the parabola described by the equation $x^2 - 20y = 0$?

Vertical (A) Signup and view all the answers

What is the vertex of the parabola defined by the equation $x^2 - 20y = 0$?

(0, 0) (D) Signup and view all the answers

What is the length of the latus rectum for a parabola with the equation $x^2 - 20y = 0$?

20 units (B) Signup and view all the answers

How far is the focus from the vertex of the parabola given by $x^2 - 20y = 0$?

5 units (D) Signup and view all the answers

What is the equation of the parabola that opens downward with vertex at $(h, k)$?

$x - h^2 = -4a(y - k)$ (C) Signup and view all the answers

Which of the following points represents an endpoint of the latus rectum for the parabola $x^2 - 20y = 0$?

(10, 5) (C) Signup and view all the answers

In which direction does the focus of the parabola $x^2 - 20y = 0$ lie in relation to the vertex?

Above the vertex (A) Signup and view all the answers

What is the equation representing a parabola that opens to the left with vertex at $(h, k)$?

$x - h^2 = -4a(y - k)$ (D) Signup and view all the answers

What is the opening direction of the parabola defined by the equation $y^2 - 8x - 4y + 20 = 0$?

Right (C) Signup and view all the answers

What is the directrix of the parabola opening to the right with vertex at $(2, 2)$?

$x = 0$ (B) Signup and view all the answers

What are the coordinates of the focus for the parabola with vertex at $(2, 2)$?

(4, 2) (B) Signup and view all the answers

What is the principal axis of symmetry for the parabola defined by $y^2 = 4ax$?

$y = k$ for some constant (D) Signup and view all the answers

What is the length of the latus rectum for the parabola with focus at $(4, 2)$?

8 units (A) Signup and view all the answers

Which of the following points are the endpoints of the latus rectum for the parabola with vertex $(2, 2)$?

(4, 6) and (4, -2) (A) Signup and view all the answers

What is the standard equation of a parabola opening to the right with vertex at the origin (0, 0)?

$y^2 = 4x$ (B) Signup and view all the answers

What value of $a$ is determined for the parabola that passes through the point (1, -2) with vertex at the origin and opening right?

$1$ (D) Signup and view all the answers

What form does the standard equation of a downward-opening parabola take?

$x - h^2 = -4a(y - k)$ (A) Signup and view all the answers

What is the distance 'a' for the parabola with vertex at (3, 1) and focus at (3, -5)?

6 (D) Signup and view all the answers

What will the equation of the parabola be if the vertex is located at (6, -3) and it opens upwards with a latus rectum length of 20?

$x^2 - 12x - 20y - 24 = 0$ (C) Signup and view all the answers

If the focus of a parabola that opens right is given as (-3, 4), what would be the equation of its principal axis?

$y = 4$ (D) Signup and view all the answers

What is the correct general equation for a parabola with focus at (-2, -3) and directrix $x = 3$?

$(y + 3)^2 = -4a(x + 2)$ (D) Signup and view all the answers

In the provided content, the vertex of the parabola with endpoints of latus rectum at (-4, 2) and (16, 2) is located at:

(6, -3) (B) Signup and view all the answers

Which statement is true regarding the parabola with the focus located at (3, -5)?

It opens downward. (C) Signup and view all the answers

What is the form of the parabolic equation for a parabola that opens to the left?

$y - k^2 = -4a(x - h)$ (D) Signup and view all the answers

Study Notes

Parabola Definition

A parabola is a conic section whose eccentricity is 𝑒 = 1.
It is defined as the locus of a point that moves in a plane so that its distance from a fixed point (Focus) is equal to its distance from a fixed line (Directrix).

Standard Form of the Equation

Equation of Parabola with Principal Axis Parallel to x-axis:
- Vertex at 𝑉(ℎ, 𝑘) and opens to the right: 𝒚−𝒌𝟐 = 𝟒𝒂 𝒙−𝒉
- Vertex at 𝑉(ℎ, 𝑘) and opens to the left: 𝒚−𝒌𝟐 = −𝟒𝒂 𝒙−𝒉
Equation of Parabola with Principal Axis Parallel to y-axis:
- Vertex at 𝑉(ℎ, 𝑘) and opens upward: 𝒙−𝒉𝟐 = 𝟒𝒂 𝒚−𝒌
- Vertex at 𝑉(ℎ, 𝑘) and opens downward: 𝒙−𝒉𝟐 = −𝟒𝒂 𝒚−𝒌

Key Components

Principal Axis/Axis of Symmetry/Axis of Parabola: Perpendicular line to the directrix passing through the focus.
Latus Rectum: The chord through the focus perpendicular to the principal axis.
Vertex: Points that cut through the principal axis.
Length of Latus Rectum: 4𝑎, where 𝑎 is the directed distance from the vertex to the focus.

Determining the Opening

Opens to the Right: If 𝑎 > 0 and the principal axis is parallel to x-axis.
Opens to the Left: If 𝑎 < 0 and the principal axis is parallel to x-axis.
Opens Upward: If 𝑎 > 0 and the principal axis is parallel to y-axis.
Opens Downward: If 𝑎 < 0 and the principal axis is parallel to y-axis.

General Equation of Parabola

The general equation of a parabola with a principal axis parallel to the y-axis is 𝑨𝒙𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎.
The general equation of a parabola with a principal axis parallel to the x-axis is 𝐁𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎.

Finding the Equation

Example: Find the equation of the parabola with vertex at the origin, axis at the x-axis, and passing through (1,-2).
Steps:
1. Determine the opening of the parabola based on the given information.
2. Use the standard equation corresponding to the opening.
3. Substitute the vertex coordinates (ℎ, 𝑘) into the standard equation.
4. Substitute the point (1, -2) into the equation and solve for 𝑎.
5. Write the final equation with the value of 𝑎.

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Description

This quiz covers the definition and standard equations of parabolas, including their key components such as the vertex, focus, and axis of symmetry. Test your knowledge on how to identify and apply the equations of parabolas based on their orientation and position.