Podcast
Questions and Answers
What is the eccentricity of a parabola?
What is the eccentricity of a parabola?
- 1 (correct)
- 2
- 0.5
- 0
Which equation represents a parabola with its principal axis parallel to the x-axis?
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?
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?
What is the standard equation of a parabola that opens upward?
What is the length of the latus rectum of a parabola?
What is the length of the latus rectum of a parabola?
Which of the following accurately describes the vertex of a parabola?
Which of the following accurately describes the vertex of a parabola?
Which parabola opens to the left based on its standard equation?
Which parabola opens to the left based on its standard equation?
Which components are essential parts of a parabola?
Which components are essential parts of a parabola?
What is the principal axis of the parabola described by the equation $x^2 - 20y = 0$?
What is the principal axis of the parabola described by the equation $x^2 - 20y = 0$?
What is the vertex of the parabola defined by the equation $x^2 - 20y = 0$?
What is the vertex of the parabola defined by the equation $x^2 - 20y = 0$?
What is the length of the latus rectum for a parabola with the equation $x^2 - 20y = 0$?
What is the length of the latus rectum for a parabola with the equation $x^2 - 20y = 0$?
How far is the focus from the vertex of the parabola given by $x^2 - 20y = 0$?
How far is the focus from the vertex of the parabola given by $x^2 - 20y = 0$?
What is the equation of the parabola that opens downward with vertex at $(h, k)$?
What is the equation of the parabola that opens downward with vertex at $(h, k)$?
Which of the following points represents an endpoint of the latus rectum for the parabola $x^2 - 20y = 0$?
Which of the following points represents an endpoint of the latus rectum for the parabola $x^2 - 20y = 0$?
In which direction does the focus of the parabola $x^2 - 20y = 0$ lie in relation to the vertex?
In which direction does the focus of the parabola $x^2 - 20y = 0$ lie in relation to the vertex?
What is the equation representing a parabola that opens to the left with vertex at $(h, k)$?
What is the equation representing a parabola that opens to the left with vertex at $(h, k)$?
What is the opening direction of the parabola defined by the equation $y^2 - 8x - 4y + 20 = 0$?
What is the opening direction of the parabola defined by the equation $y^2 - 8x - 4y + 20 = 0$?
What is the directrix of the parabola opening to the right with vertex at $(2, 2)$?
What is the directrix of the parabola opening to the right with vertex at $(2, 2)$?
What are the coordinates of the focus for the parabola with vertex at $(2, 2)$?
What are the coordinates of the focus for the parabola with vertex at $(2, 2)$?
What is the principal axis of symmetry for the parabola defined by $y^2 = 4ax$?
What is the principal axis of symmetry for the parabola defined by $y^2 = 4ax$?
What is the length of the latus rectum for the parabola with focus at $(4, 2)$?
What is the length of the latus rectum for the parabola with focus at $(4, 2)$?
Which of the following points are the endpoints of the latus rectum for the parabola with vertex $(2, 2)$?
Which of the following points are the endpoints of the latus rectum for the parabola with vertex $(2, 2)$?
What is the standard equation of a parabola opening to the right with vertex at the origin (0, 0)?
What is the standard equation of a parabola opening to the right with vertex at the origin (0, 0)?
What value of $a$ is determined for the parabola that passes through the point (1, -2) with vertex at the origin and opening right?
What value of $a$ is determined for the parabola that passes through the point (1, -2) with vertex at the origin and opening right?
What form does the standard equation of a downward-opening parabola take?
What form does the standard equation of a downward-opening parabola take?
What is the distance 'a' for the parabola with vertex at (3, 1) and focus at (3, -5)?
What is the distance 'a' for the parabola with vertex at (3, 1) and focus at (3, -5)?
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?
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?
If the focus of a parabola that opens right is given as (-3, 4), what would be the equation of its principal axis?
If the focus of a parabola that opens right is given as (-3, 4), what would be the equation of its principal axis?
What is the correct general equation for a parabola with focus at (-2, -3) and directrix $x = 3$?
What is the correct general equation for a parabola with focus at (-2, -3) and directrix $x = 3$?
In the provided content, the vertex of the parabola with endpoints of latus rectum at (-4, 2) and (16, 2) is located at:
In the provided content, the vertex of the parabola with endpoints of latus rectum at (-4, 2) and (16, 2) is located at:
Which statement is true regarding the parabola with the focus located at (3, -5)?
Which statement is true regarding the parabola with the focus located at (3, -5)?
What is the form of the parabolic equation for a parabola that opens to the left?
What is the form of the parabolic equation for a parabola that opens to the left?
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:
- Determine the opening of the parabola based on the given information.
- Use the standard equation corresponding to the opening.
- Substitute the vertex coordinates (ℎ, 𝑘) into the standard equation.
- Substitute the point (1, -2) into the equation and solve for 𝑎.
- Write the final equation with the value of 𝑎.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
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.
