Unlocking Parabolas: Graphing, Equations, and Key Features

PreAlgebra Grades High School 18:15 Video
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Lesson Description

Master parabolas! This lesson covers graphing, finding equations from key features (focus, directrix, vertex), and completing the square. Perfect for PreCalculus students.

Video Resource

Parabolas Complete Guide - Graphing and Writing Equations Using Focus, Directrix, and Vertex

Mario's Math Tutoring

Duration: 18:15
Watch on YouTube

Key Concepts

  • Parabola definition: Set of points equidistant from focus and directrix.
  • Vertex, focus, and directrix relationship.
  • Standard form equations for parabolas opening up/down and left/right.
  • Completing the square to convert general form to standard form

Learning Objectives

  • Students will be able to identify the vertex, focus, and directrix of a parabola from its equation.
  • Students will be able to graph a parabola given its vertex, focus, and directrix.
  • Students will be able to write the equation of a parabola given its vertex, focus, and directrix.
  • Students will be able to convert a quadratic equation to standard form by completing the square.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a parabola and its key components: vertex, focus, and directrix. Briefly discuss the relationship between these components. Show the video from Mario's Math Tutoring.
  • Understanding Parabola Orientation (10 mins)
    Explain how the squared variable (x or y) determines the parabola's orientation (opens up/down or left/right). Explain the role of the 'p' value in determining the direction of the parabola and the distance between the vertex and focus/directrix. Positive 'p' opens up/right, negative 'p' opens down/left.
  • Graphing Parabolas (15 mins)
    Work through examples of graphing parabolas given in standard form. Emphasize finding the vertex, focus, and directrix. Demonstrate how the '4p' value relates to the width of the parabola at the focus (focal chord). Students should practice sketching parabolas based on these features.
  • Writing Equations of Parabolas (15 mins)
    Guide students through examples of writing the equation of a parabola given its vertex and focus/directrix. Reinforce the use of the standard form equations. Discuss how to determine the sign of 'p' based on the parabola's orientation.
  • Completing the Square (15 mins)
    Demonstrate how to complete the square to convert a quadratic equation from general form to standard form. Work through several examples, emphasizing the steps involved in completing the square and rewriting the equation. Point out the importance of keeping the equation balanced.
  • Practice Problems (15 mins)
    Provide students with a set of practice problems that cover graphing parabolas, writing equations, and completing the square. Encourage students to work independently or in small groups. Circulate to provide assistance as needed.
  • Review and Wrap-up (5 mins)
    Review the key concepts covered in the lesson. Answer any remaining questions from students. Preview the next lesson topic.

Interactive Exercises

  • Parabola Matching Game
    Provide students with a set of parabola equations and a set of graphs. Have students match the equations to the corresponding graphs.
  • Equation Challenge
    Give students the vertex and either the focus or directrix of a parabola. Challenge them to find the equation and then graph it.

Discussion Questions

  • How does the value of 'p' affect the shape of the parabola?
  • Explain the relationship between the vertex, focus, and directrix of a parabola.
  • Why is it important to complete the square when the equation is not in standard form?
  • How can the axis of symmetry help you graph a parabola?

Skills Developed

  • Graphing conic sections
  • Algebraic manipulation (completing the square)
  • Problem-solving
  • Analytical skills

Multiple Choice Questions

Question 1:

The distance from the vertex to the focus of a parabola is represented by:

Correct Answer: p

Question 2:

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

Correct Answer: y is squared and p < 0

Question 3:

The vertex of the parabola (x - 2)² = 4(y + 1) is:

Correct Answer: (2, -1)

Question 4:

Which of the following equations represents a parabola opening upwards?

Correct Answer: x² = 4y

Question 5:

The directrix of a parabola is:

Correct Answer: A line equidistant from the vertex as the focus

Question 6:

What conic section is represented by the equation (y-k)^2 = 4p(x-h)?

Correct Answer: Parabola

Question 7:

Given a parabola with vertex at (0,0) and focus at (0,2), what is the equation of the directrix?

Correct Answer: y = -2

Question 8:

Which transformation is used to change the general form of a parabola to standard form?

Correct Answer: Completing the Square

Question 9:

For the equation x^2 = 8y, what is the value of 'p'?

Correct Answer: 2

Question 10:

If the focus is at (3,2) and the directrix is x=-1, what is the vertex of the parabola?

Correct Answer: (1,2)

Fill in the Blank Questions

Question 1:

The point halfway between the focus and the directrix is the __________.

Correct Answer: vertex

Question 2:

If the equation of a parabola is in the form (y - k)² = 4p(x - h), the parabola opens either to the right or to the __________.

Correct Answer: left

Question 3:

The line perpendicular to the directrix and passing through the focus is called the __________ of symmetry.

Correct Answer: axis

Question 4:

The width of a parabola at the level of the focus is equal to ___________.

Correct Answer: 4p

Question 5:

Completing the __________ is a technique used to rewrite a quadratic equation in standard form.

Correct Answer: square

Question 6:

In the standard form of a parabola equation, the variable that is squared determines the __________ of the parabola.

Correct Answer: orientation

Question 7:

The fixed point used to define a parabola is called the __________.

Correct Answer: focus

Question 8:

If p is negative and x is squared, the parabola opens __________.

Correct Answer: down

Question 9:

The standard form of a parabola that opens upwards is x^2 = __________.

Correct Answer: 4py

Question 10:

The line segment passing through the focus with endpoints on the parabola is called the __________ chord.

Correct Answer: focal

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.1.4:Write the equation of a conic section given its key features.

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