Unlocking Parabolas: Mastering Equations from Focus and Vertex

Algebra 2 Grades High School 8:32 Video
Print Lesson Plan Watch Video

Lesson Description

Learn how to derive the equation of a parabola in standard form given its focus and vertex. This lesson covers graphing techniques and understanding the relationship between the focus, vertex, and directrix.

Video Resource

Find the Equation of a Parabola Given Focus and Vertex and Graph

Mario's Math Tutoring

Duration: 8:32
Watch on YouTube

Key Concepts

  • Parabola standard form equations (vertical and horizontal)
  • Focus and vertex relationship
  • Directrix
  • Focal width (Latus Rectum)
  • Distance 'p' from vertex to focus or directrix

Learning Objectives

  • Given the focus and vertex of a parabola, students will be able to determine the standard form equation of the parabola.
  • Students will be able to graph a parabola given its equation in standard form, identifying the focus, vertex, and directrix.
  • Students will be able to determine the direction a parabola opens based on the location of the focus relative to the vertex.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general forms of parabola equations (x² = 4py and y² = 4px) and their corresponding orientations. Briefly discuss the definitions of focus, vertex, and directrix. Show the video to the class.
  • Example 1: Parabola Opening Left (10 mins)
    Work through the first example from the video (vertex at (2,-1), focus at (0,-1)). Emphasize the importance of sketching the parabola to determine the correct form of the equation. Clearly explain how to find the value of 'p' and how its sign indicates the direction of opening. Demonstrate how to use the focal width (4p) to accurately graph the parabola.
  • Example 2: Parabola Opening Up (10 mins)
    Repeat the process with the second example (vertex at (-3,2), focus at (-3,3)). Reinforce the connection between the vertex, focus, and the direction of the parabola. Show how to derive the equation and graph the parabola, including the directrix.
  • Guided Practice (15 mins)
    Have students work in pairs or small groups to solve examples 3 and 4 from the video. Circulate to provide assistance and answer questions. Encourage students to explain their reasoning and show their work.
  • Wrap-up (5 mins)
    Summarize the key steps involved in finding the equation and graphing a parabola given its focus and vertex. Emphasize the relationship between the focus, vertex, directrix, and the value of 'p'. Assign homework problems for further practice.

Interactive Exercises

  • Parabola Sketch and Solve
    Provide students with several sets of focus and vertex coordinates. Have them sketch the parabola, determine the equation, and identify the directrix.
  • Graphing Challenge
    Give students equations of parabolas in standard form. Have them identify the vertex, focus, directrix, and graph the parabola.

Discussion Questions

  • How does the location of the focus relative to the vertex determine the direction a parabola opens?
  • Explain the significance of the value 'p' in the equation of a parabola.
  • How does the focal width (latus rectum) help in graphing a parabola accurately?
  • What is the relationship between the focus and directrix and the vertex?

Skills Developed

  • Algebraic manipulation
  • Graphing quadratic functions
  • Problem-solving
  • Analytical thinking
  • Visual representation

Multiple Choice Questions

Question 1:

The distance from the vertex to the focus of a parabola is represented by what variable?

Correct Answer: p

Question 2:

If a parabola opens to the left, the value of 'p' will be:

Correct Answer: Negative

Question 3:

Which form represents a parabola opening up or down?

Correct Answer: (x-h)² = 4p(y-k)

Question 4:

The coordinates of the vertex are represented by:

Correct Answer: (h, k)

Question 5:

What is the width of the parabola at the level of the focus called?

Correct Answer: Latus Rectum

Question 6:

If the vertex of a parabola is at (1, 2) and the focus is at (1, 4), which direction does the parabola open?

Correct Answer: Up

Question 7:

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

Correct Answer: x² = 8y

Question 8:

Which of the following is true about the distance from any point on the parabola to the focus and the directrix?

Correct Answer: The distances are always equal

Question 9:

What is the equation of the directrix for the parabola (x-2)² = 4(y+1)?

Correct Answer: y = -2

Question 10:

The latus rectum is equal to?

Correct Answer: 4p

Fill in the Blank Questions

Question 1:

The line perpendicular to the axis of symmetry that does not intersect the parabola is called the _________.

Correct Answer: directrix

Question 2:

The point at which the parabola changes directions is called the _________.

Correct Answer: vertex

Question 3:

The line through the focus that is perpendicular to the directrix is the _________.

Correct Answer: axis of symmetry

Question 4:

The formula for the focal width of a parabola is _________.

Correct Answer: 4p

Question 5:

If the vertex is at (0,0) and the focus is at (3,0), then p= _________.

Correct Answer: 3

Question 6:

For a parabola opening downwards, the coefficient of the y term will be _________.

Correct Answer: negative

Question 7:

The general form of a parabola opening right or left is y - k squared equals _________.

Correct Answer: 4p(x-h)

Question 8:

The segment that goes through the focus and has endpoints on the parabola is called _________.

Correct Answer: focal chord

Question 9:

If the focus is located at (2,5) and the vertex at (2,3), the directrix would be the line _________.

Correct Answer: y=1

Question 10:

The distance 'p' is the same from the vertex to the focus as it is to the _________.

Correct Answer: directrix

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.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans