Unlocking Parabolas: Finding Equations from Focus and Vertex

Algebra 2 Grades High School 2:23 Video
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Lesson Description

Learn how to determine the equation of a parabola given its focus and vertex. This lesson uses a visual approach and clear explanations to help you master this key Algebra 2 concept.

Video Resource

Finding Equation of Parabola Given Focus and Vertex

Mario's Math Tutoring

Duration: 2:23
Watch on YouTube

Key Concepts

  • Parabola definition and properties (focus, vertex, directrix)
  • Focal distance (P)
  • Standard equations of parabolas (x² = 4py and y² = 4px)
  • Relationship between the sign of 'P' and the direction of the parabola's opening

Learning Objectives

  • Students will be able to define the key components of a parabola: vertex, focus, and directrix.
  • Students will be able to determine the equation of a parabola given its vertex and focus.
  • Students will be able to sketch a parabola based on its vertex, focus, and calculated equation.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a parabola, its vertex, focus, and directrix. Briefly discuss the concept of focal distance (P) and its significance. Show the video to the class.
  • Video Analysis (10 mins)
    After watching the video, discuss the example problem. Emphasize the importance of sketching the parabola to determine its orientation (up, down, left, or right). Review the two standard forms of the parabola equation (x² = 4py and y² = 4px) and how the sign of 'P' affects the parabola's direction.
  • Guided Practice (15 mins)
    Work through several example problems together as a class. Vary the location of the vertex (e.g., (0,0), (2,3), (-1, -4)) and the focus to cover different scenarios. Guide students through the steps: sketch the parabola, determine the direction it opens, identify the correct equation form, calculate 'P', and write the equation.
  • Independent Practice (15 mins)
    Assign a set of practice problems for students to work on individually. Circulate the classroom to provide assistance and answer questions. Problems should progressively increase in difficulty.
  • Wrap-up (5 mins)
    Review the key concepts and address any remaining questions. Preview the next lesson on parabolas (e.g., finding the focus and vertex from the equation).

Interactive Exercises

  • Parabola Sketching Game
    Provide students with different vertex and focus coordinates. Have them compete to sketch the parabola and determine its equation the fastest. This can be done on whiteboards or using online graphing tools.
  • Equation Matching
    Create a set of cards with parabola equations and a set of cards with corresponding vertex and focus coordinates. Students must match the correct equation to the given vertex and focus.

Discussion Questions

  • How does the distance from any point on the parabola to the focus relate to the distance from that point to the directrix?
  • What are the key differences between the equations x² = 4py and y² = 4px, and how do they relate to the orientation of the parabola?
  • How does the location of the vertex impact the equation of the parabola?

Skills Developed

  • Problem-solving
  • Visual reasoning
  • Algebraic manipulation
  • Analytical thinking

Multiple Choice Questions

Question 1:

The vertex of a parabola is located halfway between which two points/lines?

Correct Answer: Focus and directrix

Question 2:

What does the variable 'P' represent in the standard equation of a parabola?

Correct Answer: Distance from the vertex to the focus

Question 3:

If a parabola opens downward, which of the following is true regarding the value of P?

Correct Answer: P is negative

Question 4:

Which equation represents a parabola that opens to the right?

Correct Answer: y² = 4px

Question 5:

A parabola has a vertex at (0,0) and a focus at (0,3). What is the value of P?

Correct Answer: 3

Question 6:

A parabola has a vertex at (0,0) and a focus at (0,-2). What is the equation of the parabola?

Correct Answer: x² = -8y

Question 7:

What is the equation of a parabola with vertex (0,0) and focus at (2,0)?

Correct Answer: y² = 8x

Question 8:

The equation of a parabola is x² = -12y. What is the value of 'P'?

Correct Answer: -3

Question 9:

What is the definition of a parabola?

Correct Answer: A parabola is a two-dimensional, mirror-symmetrical curve

Question 10:

How can you find the equation of a parabola?

Correct Answer: Find the vertex and focus

Fill in the Blank Questions

Question 1:

The point where a parabola changes direction is called the ______.

Correct Answer: vertex

Question 2:

The line that is equidistant from all points on a parabola is the ______.

Correct Answer: directrix

Question 3:

The distance from the vertex to the focus is called the ______ distance.

Correct Answer: focal

Question 4:

If the equation of a parabola is y² = 4px and P is positive, the parabola opens to the ______.

Correct Answer: right

Question 5:

If the equation of a parabola is x² = 4py and P is negative, the parabola opens ______.

Correct Answer: downward

Question 6:

For a parabola with vertex (0,0) and focus (0,5), the value of P is ______.

Correct Answer: 5

Question 7:

The equation x² = -20y represents a parabola opening ______.

Correct Answer: downward

Question 8:

For the parabola y² = 16x, the value of P is ______.

Correct Answer: 4

Question 9:

The two forms of an equation are x² = 4py and ______.

Correct Answer: y² = 4px

Question 10:

What is an essential component needed to find the equation of a parabola?

Correct Answer: focus and vertex

Educational Standards

A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.F.1.3:Graph a quadratic function. Identify the domain, range, x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology.

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