Unlocking Parabolas: Vertex, Directrix, and Equations

Algebra 2 Grades High School 4:43 Video
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Lesson Description

Master the art of writing parabola equations when given the vertex and directrix. This lesson explores the relationship between these key features and their impact on the parabola's equation and graph.

Video Resource

Write Equation of a Parabola Given Vertex and Directrix

Mario's Math Tutoring

Duration: 4:43
Watch on YouTube

Key Concepts

  • Parabola definition (focus, directrix, vertex)
  • Standard form equation of a parabola
  • Relationship between vertex, directrix, and the 'p' value

Learning Objectives

  • Students will be able to define a parabola in terms of its vertex, focus, and directrix.
  • Students will be able to determine the orientation (up, down, left, right) of a parabola based on the position of the directrix relative to the vertex.
  • Students will be able to write the equation of a parabola given its vertex and directrix.
  • Students will be able to graph a parabola given its vertex, directrix, and equation.

Educator Instructions

  • Introduction (5 mins)
    Begin with a brief review of quadratic functions and their graphs. Introduce the concept of a parabola as a special type of quadratic function. Show a few real-world examples of parabolas (satellite dishes, suspension bridges).
  • Parabola Definition and Key Features (10 mins)
    Explain the definition of a parabola in terms of the focus and directrix. Define the vertex as the point where the parabola bends. Explain the 'p' value as the distance between the vertex and the focus, and the vertex and the directrix.
  • Equation of a Parabola (15 mins)
    Introduce the standard form equations of a parabola: x² = 4py and y² = 4px. Explain how the sign of 'p' determines the direction the parabola opens (up/down or left/right). Show how to shift the graph horizontally and vertically using the vertex form: (x-h)² = 4p(y-k) and (y-k)² = 4p(x-h).
  • Example Problem (15 mins)
    Work through the example problem from the video: Vertex (4, -2), Directrix y = 2. Guide students through the steps: 1) Plot the vertex and directrix. 2) Determine the direction the parabola opens (down). 3) Calculate the 'p' value (distance from vertex to directrix). 4) Substitute the vertex (h, k) and 'p' value into the correct standard form equation. 5) Discuss how the '4p' value relates to the width of the parabola, including how to determine the width at the focus.
  • Practice Problems (15 mins)
    Provide students with several practice problems where they are given the vertex and directrix and asked to write the equation of the parabola. Encourage them to graph the parabola to visualize the solution.
  • Wrap-up (5 mins)
    Summarize the key concepts and address any remaining questions. Preview the next lesson, which could involve finding the equation of a parabola given the focus and vertex.

Interactive Exercises

  • Graphing Parabolas with GeoGebra
    Use GeoGebra or a similar online graphing tool to graph parabolas given their equations. Students can manipulate the 'h', 'k', and 'p' values to see how they affect the parabola's position and shape.
  • Parabola Matching Game
    Create a matching game where students match equations of parabolas to their corresponding graphs, vertices, and directrices.

Discussion Questions

  • How does changing the 'p' value affect the shape of the parabola?
  • What is the relationship between the vertex and the axis of symmetry of a parabola?
  • How can you tell from the equation whether a parabola opens up/down or left/right?
  • Why is the distance from any point on the parabola to the focus equal to the distance from that point to the directrix?

Skills Developed

  • Algebraic manipulation
  • Graphing skills
  • Problem-solving
  • Analytical thinking

Multiple Choice Questions

Question 1:

A parabola is defined as the set of all points equidistant from a point called the _______ and a line called the _______.

Correct Answer: focus, directrix

Question 2:

If the directrix of a parabola is y = 5 and the vertex is (2, 1), in what direction does the parabola open?

Correct Answer: Down

Question 3:

The distance between the vertex and the focus of a parabola is represented by the variable:

Correct Answer: p

Question 4:

If a parabola opens to the left, which of the following is true about its equation?

Correct Answer: The y term is squared and p is negative

Question 5:

What is the vertex of the parabola represented by the equation (x - 3)² = 8(y + 2)?

Correct Answer: (3, -2)

Question 6:

Which of the following equations represents a parabola that opens upwards?

Correct Answer: (x + 2)² = 4(y - 1)

Question 7:

If the vertex of a parabola is at (0, 0) and p = -2, and the parabola opens downward, what is the equation of the parabola?

Correct Answer: x² = -8y

Question 8:

The width of the parabola at the level of the focus is equal to:

Correct Answer: 4p

Question 9:

Given the equation (y - 1)² = 12(x + 2), what is the value of 'p'?

Correct Answer: 24

Question 10:

The 'h' and 'k' in the general form of a parabola's equation (x-h)² = 4p(y-k) represent:

Correct Answer: The vertex

Fill in the Blank Questions

Question 1:

The point where the parabola 'bends' is called the _________.

Correct Answer: vertex

Question 2:

A parabola opening to the right has a _______ 'p' value.

Correct Answer: positive

Question 3:

The line y = 3 is a _________ line.

Correct Answer: horizontal

Question 4:

The distance from the vertex to the directrix is the same as the distance from the vertex to the __________.

Correct Answer: focus

Question 5:

If the vertex is at (2, -1), then h = _______ and k = _______.

Correct Answer: 2,-1

Question 6:

For a parabola opening downwards, the 'p' value is __________.

Correct Answer: negative

Question 7:

The general form of a parabola equation that opens up or down is (x - h)² = 4p(y - k). The general form of a parabola equation that opens right or left is __________.

Correct Answer: (y - k)² = 4p(x - h)

Question 8:

If the focus is at (0,3) and the vertex is at (0,0), then the p value is __________.

Correct Answer: 3

Question 9:

Given (x + 5)² = -8(y - 2), the value of 4p is __________.

Correct Answer: -8

Question 10:

At the level of the focus, the width of the parabola is equal to __________.

Correct Answer: 4p

Educational Standards

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. 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.

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