Unlocking Parabolas: Mastering Equations from Focus and Directrix

Algebra 2 Grades High School 3:43 Video
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Lesson Description

Learn how to derive the equation of a parabola given its focus and directrix. This lesson covers the fundamental definition, formula, and application through a step-by-step example.

Video Resource

Equation of Parabola Given Focus and Directrix

Mario's Math Tutoring

Duration: 3:43
Watch on YouTube

Key Concepts

  • Definition of a parabola: set of points equidistant to the focus and directrix.
  • Understanding the formula for a parabola in vertex form.
  • Determining the direction a parabola opens based on the equation and the position of focus and directrix.
  • The 'p' value represents the distance from the vertex to the focus and from the vertex to the directrix.

Learning Objectives

  • Students will be able to define a parabola in terms of its focus and directrix.
  • Students will be able to determine the equation of a parabola given its focus and directrix.
  • Students will be able to graph a parabola given its equation, focus, and directrix.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a parabola, focus, and directrix. Discuss the key properties of parabolas and their real-world applications (e.g., satellite dishes, reflectors).
  • Video Presentation (10 mins)
    Play the Mario's Math Tutoring video 'Equation of Parabola Given Focus and Directrix'. Encourage students to take notes on the formula, how to determine the direction of the parabola, and the steps for finding the equation.
  • Guided Practice (15 mins)
    Work through the example problem from the video (Directrix x=-1 and Focus (3,1)) step-by-step, clarifying each step. Emphasize the importance of finding the vertex and the value of 'p'. Then demonstrate the calculation of how wide the parabola is at the level of the focus (4p).
  • Independent Practice (15 mins)
    Provide students with practice problems where they need to find the equation of a parabola given the focus and directrix. Include problems where the parabola opens up, down, left, and right.
  • Wrap-up and Assessment (5 mins)
    Summarize the key concepts of the lesson. Assign the multiple choice and fill in the blank quizzes for assessment.

Interactive Exercises

  • Graphing Activity
    Have students use graphing paper or a graphing calculator to graph parabolas given their equations. Students should identify the focus, directrix, and vertex on their graphs. Then, change the value of 'p' in the equation and observe how it affects the graph.
  • Directrix and Focus Placement
    Give students a set of coordinate planes. For each coordinate plane, give students coordinates for a focus and an equation for a directrix. Students will calculate the 'p' value and identify the equation of the parabola for the set of focus and directrix given. Have students share answers and explain steps on the board.

Discussion Questions

  • How does the location of the focus and directrix affect the shape of the parabola?
  • Explain the significance of the 'p' value in the equation of a parabola.
  • Can you think of real-world examples where parabolas are used, and how the focus and directrix are important in those applications?

Skills Developed

  • Problem-solving
  • Analytical thinking
  • Equation derivation
  • Graphing skills

Multiple Choice Questions

Question 1:

What defines a parabola?

Correct Answer: The set of all points equidistant from a point (focus) and a line (directrix).

Question 2:

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

Correct Answer: p

Question 3:

If a parabola has the equation (y-k)^2 = 4p(x-h), and 'p' is positive, which direction does the parabola open?

Correct Answer: Right

Question 4:

What is the vertex of a parabola?

Correct Answer: The midpoint between the focus and the directrix.

Question 5:

Given a focus at (0, 2) and a directrix at y = -2, what is the value of 'p'?

Correct Answer: 2

Question 6:

Which of the following equation types represents a parabola that opens either up or down?

Correct Answer: (x - h)^2 = 4p(y - k)

Question 7:

How wide is a parabola at the level of the focus?

Correct Answer: 4p

Question 8:

The directrix of a parabola is...

Correct Answer: A line outside the parabola

Question 9:

What are the coordinates of the vertex given the equation (x-3)^2 = 8(y+2)?

Correct Answer: (3, -2)

Question 10:

If a parabola opens to the left, what must be true about the value of 'p' in the equation (y-k)^2 = 4p(x-h)?

Correct Answer: p < 0

Fill in the Blank Questions

Question 1:

The __________ is the point inside the parabola that is used in the definition.

Correct Answer: focus

Question 2:

The line outside the parabola used in the definition is called the __________.

Correct Answer: directrix

Question 3:

The turning point of a parabola is called the __________.

Correct Answer: vertex

Question 4:

If the equation of a parabola is x^2 = 4py, and p is positive, the parabola opens ___________.

Correct Answer: up

Question 5:

The distance from the vertex to the focus is equal to the distance from the vertex to the __________.

Correct Answer: directrix

Question 6:

In the equation (y-k)^2 = 4p(x-h), the coordinates of the vertex are represented by (____, ____).

Correct Answer: h, k

Question 7:

If the focus of a parabola is at (0, -3) and the vertex is at (0, 0), the value of 'p' is ____.

Correct Answer: -3

Question 8:

A parabola that opens to the right has a _______ squared term.

Correct Answer: y

Question 9:

The axis of _______ of a parabola passes through the focus and vertex.

Correct Answer: symmetry

Question 10:

At the level of the focus, a parabola is ______ wide.

Correct Answer: 4p

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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