Unlocking Parabolas: Focus, Vertex, and Equation

Algebra 2 Grades High School 10:02 Video
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Lesson Description

Learn how to determine the equation of a parabola when given its focus and vertex. This lesson covers identifying the orientation of the parabola and applying the correct formula.

Video Resource

Given Focus and Vertex Find Equation

Kevinmathscience

Duration: 10:02
Watch on YouTube

Key Concepts

  • Parabola definition and properties
  • Focus and vertex of a parabola
  • Directrix of a parabola
  • Standard equation of a parabola
  • Determining parabola orientation (up, down, left, right)

Learning Objectives

  • Students will be able to identify the vertex and focus of a parabola from a given graph or coordinates.
  • Students will be able to determine the orientation (up, down, left, or right) of a parabola given its focus and vertex.
  • Students will be able to derive the equation of a parabola given its focus and vertex.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a parabola, its key features (vertex, focus, directrix), and the two standard forms of the parabola equation: (x-h)^2 = 4p(y-k) and (y-k)^2 = 4p(x-h). Emphasize that understanding these equations is crucial for the lesson.
  • Video Viewing (10 mins)
    Play the "Given Focus and Vertex Find Equation" video by Kevinmathscience. Instruct students to take notes on the steps involved in determining the equation of a parabola.
  • Guided Practice (15 mins)
    Work through examples similar to those in the video, emphasizing the importance of plotting the focus and vertex to determine the parabola's orientation. Guide students through identifying the values of h, k, and p, and substituting them into the correct standard equation. Provide positive and negative values for all variables.
  • Independent Practice (15 mins)
    Assign practice problems where students determine the equation of a parabola given its focus and vertex. Circulate to provide assistance and answer questions. Problems should include examples of each orientation (up, down, left, right) and various placements on the coordinate plane.
  • Review and Assessment (5 mins)
    Review the key steps and concepts covered in the lesson. Administer the multiple choice and fill-in-the-blank quizzes to assess student understanding.

Interactive Exercises

  • Graphing Parabolas
    Students use graphing paper or a graphing calculator to plot the focus, vertex, and directrix of several parabolas. Then, they sketch the parabola and determine its equation.
  • Equation Matching
    Provide students with a list of parabolas (described by focus and vertex) and a list of equations. Students must match each parabola description to its correct equation.

Discussion Questions

  • How does the location of the focus relative to the vertex determine the parabola's orientation?
  • What is the significance of the 'p' value in the standard equation of a parabola?
  • Explain why it is important to plot the vertex and focus before attempting to calculate the equation.

Skills Developed

  • Analytical skills
  • Problem-solving skills
  • Algebraic manipulation
  • Visual-spatial reasoning

Multiple Choice Questions

Question 1:

The vertex of a parabola is (1, -2) and the focus is (1, 0). Which direction does the parabola open?

Correct Answer: Up

Question 2:

What value does 'p' represent in the standard equation of a parabola?

Correct Answer: The distance from the vertex to the directrix.

Question 3:

Which equation represents a parabola that opens to the left?

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

Question 4:

If the vertex is (2, 3) what values are h and k?

Correct Answer: h = 2, k = 3

Question 5:

The focus of a parabola is (0, -3) and the vertex is (0, 0). What is the value of 'p'?

Correct Answer: -3

Question 6:

The equation of a parabola is (x - 1)^2 = 8(y + 2). What is the vertex of the parabola?

Correct Answer: (1, -2)

Question 7:

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

Correct Answer: Negative

Question 8:

What is the directrix of a parabola?

Correct Answer: A line outside the parabola that is used to define it

Question 9:

Which is the correct equation to use for a parabola with vertex (h,k) opening up or down?

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

Question 10:

How many units away from the vertex is the directrix?

Correct Answer: p

Fill in the Blank Questions

Question 1:

The point inside the curve of the parabola is known as the ________.

Correct Answer: focus

Question 2:

The line x = -2 is a __________.

Correct Answer: directrix

Question 3:

The formula for finding the equation when the y is squared is (y-k)^2 = _______(x-h).

Correct Answer: 4p

Question 4:

The vertex represents what point of the parabola when opening up or down?

Correct Answer: minimum

Question 5:

If the focus is above the vertex, the parabola opens _______.

Correct Answer: up

Question 6:

If p = 4, then 4p = _______.

Correct Answer: 16

Question 7:

The general formula (x - h)^2 = 4p(y - k) means that (h, k) represents the _______.

Correct Answer: vertex

Question 8:

If a parabola opens to the right, the equation will have the _______ term squared.

Correct Answer: y

Question 9:

The distance from the vertex to the focus is represented by the variable ______.

Correct Answer: p

Question 10:

For a parabola that opens downward, the value of 'p' is always ________.

Correct Answer: negative

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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