Unlocking Parabolas: Finding Equations from Vertex and Directrix

Algebra 2 Grades High School 2:00 Video
Print Lesson Plan Watch Video

Lesson Description

Learn to determine the equation of a parabola when given its vertex and directrix. This lesson utilizes geometric properties and algebraic manipulation to derive the standard form equation.

Video Resource

Finding Equation of Parabola Given Vertex and Directrix

Mario's Math Tutoring

Duration: 2:00
Watch on YouTube

Key Concepts

  • Parabola Definition: A U-shaped graph with a vertex, focus, and directrix.
  • Vertex-Directrix Relationship: The vertex is equidistant from the focus and the directrix.
  • Focal Distance (p): The distance from the vertex to the focus (or vertex to the directrix).
  • Standard Forms: Recognizing and applying the standard forms of parabolic equations (x² = 4py or y² = 4px) and their shifted forms.
  • Transformations: Understanding how horizontal and vertical shifts affect the equation of the parabola.

Learning Objectives

  • Students will be able to identify the vertex and directrix of a parabola from given information.
  • Students will be able to calculate the focal distance (p) using the vertex and directrix.
  • Students will be able to determine the correct standard form of a parabola equation (x² or y² term) based on its orientation.
  • Students will be able to write the equation of a parabola given its vertex and directrix.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a parabola, its key components (vertex, focus, directrix), and the concept of focal distance. Briefly discuss the two standard forms of parabolic equations: x² = 4py and y² = 4px.
  • Video Viewing and Note-Taking (10 mins)
    Play the 'Finding Equation of Parabola Given Vertex and Directrix' video by Mario's Math Tutoring. Instruct students to take notes on the key steps and concepts explained in the video. Encourage them to pause the video as needed to clarify any confusion.
  • Guided Example (15 mins)
    Work through the example provided in the video step-by-step, emphasizing the geometric interpretation of the vertex and directrix. Highlight the importance of determining the correct standard form and calculating the focal distance. Discuss the transformations (horizontal and vertical shifts) and how they affect the equation.
  • Practice Problems (15 mins)
    Present students with additional practice problems involving different vertex and directrix locations. Encourage them to work individually or in pairs to find the equations of the parabolas. Provide guidance and support as needed.
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts and steps involved in finding the equation of a parabola given its vertex and directrix. Answer any remaining questions from students. Preview upcoming topics related to conic sections.

Interactive Exercises

  • Graphing Tool Activity
    Use an online graphing tool (e.g., Desmos) to graph parabolas based on given equations. Explore how changing the parameters (vertex coordinates, focal distance) affects the shape and position of the parabola.
  • Vertex and Directrix Challenge
    Present students with a set of vertex and directrix coordinates. Have them compete to see who can correctly determine the equation of the parabola the fastest. This can be done individually or in teams.

Discussion Questions

  • How does the location of the directrix relative to the vertex determine whether the parabola opens up/down or left/right?
  • What is the significance of the focal distance (p) in determining the shape of the parabola?
  • How do horizontal and vertical shifts affect the standard equation of the parabola?

Skills Developed

  • Geometric Reasoning
  • Algebraic Manipulation
  • Problem-Solving
  • Visual Representation

Multiple Choice Questions

Question 1:

The vertex of a parabola is located at (3, -2) and its directrix is y = 1. What is the value of 'p', the focal distance?

Correct Answer: -3

Question 2:

A parabola opens downward. Which of the following is true about the 'p' value?

Correct Answer: p < 0

Question 3:

If the vertex of a parabola is (h, k), what do 'h' and 'k' represent in the shifted standard equation?

Correct Answer: Horizontal and vertical shifts

Question 4:

The directrix of a parabola is a ______.

Correct Answer: Line

Question 5:

Which form of the equation indicates a parabola that opens either left or right?

Correct Answer: y² = 4px

Question 6:

A parabola has a vertex at (-1, 4) and opens upwards. What is the sign of the 'p' value?

Correct Answer: Positive

Question 7:

The distance from the vertex to the focus of a parabola is always ____ the distance from the vertex to the directrix.

Correct Answer: Equal to

Question 8:

What is the standard form equation of a parabola opening upwards with a vertex at the origin?

Correct Answer: x² = 4py

Question 9:

Given a vertex and directrix, what is the first step to finding the parabola's equation?

Correct Answer: Determine the orientation and find 'p'

Question 10:

A parabola opens to the left. What can you conclude?

Correct Answer: The 'p' value is negative and the equation is in the form y² = 4px

Fill in the Blank Questions

Question 1:

The __________ of a parabola is the point where the parabola changes direction.

Correct Answer: vertex

Question 2:

The distance between the vertex and the focus (or vertex and directrix) is called the __________ __________.

Correct Answer: focal distance

Question 3:

If a parabola opens downward, the 'p' value will be __________.

Correct Answer: negative

Question 4:

The standard equation of a parabola that opens to the right is y² = __________.

Correct Answer: 4px

Question 5:

The directrix is a __________ that does not intersect the parabola.

Correct Answer: line

Question 6:

Horizontal and vertical shifts are represented by 'h' and 'k' in the __________ equation of the parabola.

Correct Answer: shifted

Question 7:

If the equation contains x², the parabola opens either __________ or __________.

Correct Answer: up/down

Question 8:

The focus of a parabola is a __________ inside the curve of the parabola.

Correct Answer: point

Question 9:

To find the equation, first determine orientation and the value of __________.

Correct Answer: p

Question 10:

A y² term in a parabolic equation indicates that the parabola opens either left or __________.

Correct Answer: right

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