Unlocking Parabolas: Vertex, Focus, Directrix, and Graphing

PreAlgebra Grades High School 8:33 Video
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Lesson Description

Master the art of analyzing and graphing parabolas by understanding their standard forms, vertices, foci, and directrices. Learn to complete the square and extract key information for accurate sketching.

Video Resource

Parabola Find Vertex, Focus, Directrix, and Graph

Mario's Math Tutoring

Duration: 8:33
Watch on YouTube

Key Concepts

  • Standard form of a parabola
  • Vertex, focus, and directrix of a parabola
  • Completing the square
  • Relationship between the equation and the graph of a parabola
  • P-value as the distance from the vertex to focus or directrix

Learning Objectives

  • Rewrite a parabolic equation into standard form by completing the square.
  • Identify the vertex, focus, and directrix of a parabola given its equation.
  • Sketch the graph of a parabola using its vertex, focus, and directrix.
  • Determine the direction a parabola opens (up, down, left, right) based on its equation.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definition of a parabola as the set of all points equidistant from a focus and a directrix. Introduce the two standard forms of a parabola (y² = 4px and x² = 4py). Explain the importance of identifying the vertex, focus, and directrix for graphing.
  • Example 1: y² Type Parabola (15 mins)
    Present the first example from the video. Guide students through the process of completing the square to rewrite the equation in standard form. Emphasize the steps of moving terms, completing the square, factoring, and identifying the vertex. Show how to determine the value of 'p' and use it to find the focus and directrix. Graph the parabola, labeling the vertex, focus, and directrix.
  • Example 2: x² Type Parabola (15 mins)
    Present the second example from the video. Walk through the steps of rearranging the equation, completing the square, and factoring. Explain how the negative sign in the equation affects the direction the parabola opens. Identify the vertex, focus, and directrix, and use them to sketch the graph. Highlight the concept of the focal chord.
  • Practice and Discussion (10 mins)
    Provide students with additional practice problems of varying difficulty levels. Facilitate a class discussion to address any questions or misconceptions. Encourage students to share their strategies for solving these types of problems.

Interactive Exercises

  • Completing the Square Practice
    Provide students with several parabolic equations and have them practice completing the square to rewrite them in standard form. Have students work in groups and check each other's work.
  • Graphing Challenge
    Give students a set of parabolas (equations) and have them work in pairs to identify the vertex, focus, directrix and graph them. The pair who gets the most correct in a time limit wins.

Discussion Questions

  • How does the sign of 'p' affect the direction the parabola opens?
  • What is the relationship between the vertex, focus, and directrix of a parabola?
  • Why is it important to complete the square before identifying the key features of a parabola?
  • How can the focal chord help you sketch an accurate graph of a parabola?

Skills Developed

  • Algebraic manipulation (completing the square)
  • Analytical skills (identifying key features of parabolas)
  • Graphing skills
  • Problem-solving
  • Critical thinking

Multiple Choice Questions

Question 1:

The standard form of a parabola with a vertical axis of symmetry is given by:

Correct Answer: (x-h)² = 4p(y-k)

Question 2:

The vertex of the parabola (x-2)² = 8(y+1) is:

Correct Answer: (2, -1)

Question 3:

In the equation (y+3)² = -4(x-1), what is the value of 'p'?

Correct Answer: -1

Question 4:

If the vertex of a parabola is at (0,0) and the focus is at (0,2), the directrix is the line:

Correct Answer: y = -2

Question 5:

For a parabola that opens to the left, which of the following is true?

Correct Answer: p < 0 and y² term

Question 6:

What is the first step in rewriting y² + 6y - 8x + 1 = 0 in standard form?

Correct Answer: Move the x and constant terms to the right side

Question 7:

The distance between the vertex and the focus of a parabola is:

Correct Answer: |p|

Question 8:

The focal chord is how wide at the level of the focus?

Correct Answer: 4p

Question 9:

The directrix is a _______.

Correct Answer: Line

Question 10:

For the equation (x + 5)^2 = 12(y - 2), what is the y-coordinate of the focus?

Correct Answer: 5

Fill in the Blank Questions

Question 1:

The point where the parabola changes direction is called the ________.

Correct Answer: vertex

Question 2:

The line that is equidistant from the vertex as the focus is called the ________.

Correct Answer: directrix

Question 3:

Completing the ________ is a method used to rewrite a quadratic expression in a more convenient form.

Correct Answer: square

Question 4:

The standard form of a parabola opening left or right has a squared ______ variable.

Correct Answer: y

Question 5:

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

Correct Answer: p

Question 6:

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

Correct Answer: down

Question 7:

The focus always lies on the ________ of the parabola.

Correct Answer: axis

Question 8:

The equation of a horizontal line is always in the form y = ________.

Correct Answer: constant

Question 9:

The length of the focal chord is always equal to _______.

Correct Answer: 4p

Question 10:

When completing the square for y² + 4y, you need to add ________ to both sides of the equation.

Correct Answer: 4

Educational Standards

PC.CS.1.2:Identify key features of conic sections (foci, directrix, radii, axes, asymptotes, center) graphically and algebraically. PC.CS.1.3:Sketch a graph of a conic section using its key features. PC.CS.1.5:Given the equation 𝑎𝑥² + 𝑏𝑦² + 𝑐𝑥 + 𝑑𝑦 + 𝑒 = 0, determine if the equation represents a circle, ellipse, parabola, or hyperbola. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation.

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