Beyond the Origin: Mastering Parabolas with Directrix and Focus

Algebra 2 Grades High School 19:28 Video
Print Lesson Plan Watch Video

Lesson Description

Learn to graph parabolas not centered at the origin using their directrix and focus. This lesson builds on prior knowledge of parabolas and introduces transformations.

Video Resource

Parabola Directrix Focus | Part 2

Kevinmathscience

Duration: 19:28
Watch on YouTube

Key Concepts

  • Vertex form of a parabola equation
  • Focus and directrix of a parabola
  • Transformations of parabolas (horizontal and vertical shifts)
  • Relationship between 'p' value, focus, and directrix

Learning Objectives

  • Students will be able to identify the vertex, focus, and directrix of a parabola given its equation in vertex form.
  • Students will be able to graph parabolas not centered at the origin using the focus, directrix, and vertex.
  • Students will be able to determine the equation of a parabola given its vertex, focus, and directrix.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concepts of parabolas, focus, and directrix from the previous lesson. Emphasize that this lesson will focus on parabolas where the vertex is NOT at the origin.
  • Video Viewing (15 mins)
    Play the Kevinmathscience video 'Parabola Directrix Focus | Part 2'. Encourage students to take notes on the formulas for parabolas with vertices not at the origin and the steps for graphing.
  • Guided Practice (20 mins)
    Work through example problems similar to those in the video. Start with simpler examples and gradually increase the complexity. Emphasize the importance of correctly identifying 'h', 'k', and 'p' values. Explicitly show how these values relate to the shifts from the origin.
  • Independent Practice (15 mins)
    Assign practice problems where students graph parabolas and determine the equations of parabolas given the focus, directrix, and vertex. This can be done individually or in small groups.
  • Wrap-up and Assessment (5 mins)
    Summarize the key concepts and address any remaining questions. Prepare students for the quiz and/or homework assignment.

Interactive Exercises

  • Desmos Graphing Challenge
    Students use Desmos (or a similar graphing tool) to graph parabolas given equations in vertex form. Then, they can change the 'h', 'k', and 'p' values to observe how the graph changes dynamically.
  • Matching Activity
    Create a matching activity where students match parabola equations with their corresponding graphs, vertex coordinates, focus coordinates, and directrix equations.

Discussion Questions

  • How do the values of 'h' and 'k' in the vertex form of a parabola equation affect the graph?
  • Explain the relationship between the vertex, focus, and directrix of a parabola. How does the 'p' value relate to these elements?
  • What are some real-world applications of parabolas?

Skills Developed

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

Multiple Choice Questions

Question 1:

The vertex of the parabola given by the equation (x - 3)² = 4(y + 2) is at what point?

Correct Answer: (3, -2)

Question 2:

For a parabola, the distance from the vertex to the focus is represented by:

Correct Answer: p

Question 3:

If a parabola opens upwards, which of the following is true about the coefficient 'p' in its equation?

Correct Answer: p > 0

Question 4:

The equation of the directrix for the parabola (x + 1)² = -8(y - 3) is:

Correct Answer: y = 5

Question 5:

The focus of the parabola (y - 1)² = 12(x + 2) is at what point?

Correct Answer: (1, -2)

Question 6:

What does 'h' represent in the vertex form of a parabola equation?

Correct Answer: Horizontal shift

Question 7:

A parabola facing left will have which variable squared?

Correct Answer: y

Question 8:

If the vertex of a parabola is at (2, -1) and p = 3, and the parabola opens upward, what is the y-coordinate of the focus?

Correct Answer: 2

Question 9:

The distance across the parabola through the focus is equal to which of the following?

Correct Answer: 4p

Question 10:

What is the equation for the Directrix of a parabola?

Correct Answer: All of the above

Fill in the Blank Questions

Question 1:

In the vertex form of a parabola, (x - h)² = 4p(y - k), the point (h, k) represents the __________.

Correct Answer: vertex

Question 2:

The line that is equidistant from the vertex and the focus of a parabola is called the __________.

Correct Answer: directrix

Question 3:

The value of 'p' represents the distance from the vertex to the _______ and from the vertex to the ________.

Correct Answer: focus

Question 4:

If a parabola opens to the right, its equation will have the form (y - k)² = 4p(x - h), where p is _______.

Correct Answer: positive

Question 5:

If the equation of a parabola is (x + 2)² = -8(y - 1), the vertex is at the point (______, ______)

Correct Answer: -2

Question 6:

If the equation of a parabola is (x + 2)² = -8(y - 1), the vertex is at the point (-2, ______)

Correct Answer: 1

Question 7:

The distance across the parabola through the focus is called the ________ ________.

Correct Answer: focal width

Question 8:

If the vertex of a parabola is (0, 0) and the focus is (0, 2), the directrix is the line y = _______.

Correct Answer: -2

Question 9:

For the equation (y + 3)² = 4(x - 1), the parabola opens to the _______.

Correct Answer: right

Question 10:

Given the equation (x - h)² = 4p(y - k), if p is negative the parabola opens ________.

Correct Answer: downward

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.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically. 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.

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