Lesson Plan: Quadratic Functions and Equations: Conic Sections

Lesson Summary

This 50-minute lesson explores the parabola as a conic section, focusing on its geometric properties and derivation. Students will learn to describe parabolas using focus and directrix, and graph them accordingly. The lesson incorporates multimedia resources from Media4Math.com to enhance understanding. Activities include identifying conic sections, deriving and graphing parabolas, and exploring real-world applications. A 10-question quiz with an answer key is provided for assessment.

Lesson Objectives

By the end of this lesson, students will be able to:

1. Describe the parabola as a conic section and its geometric properties
2. Derive and graph parabolas using focus and directrix
3. Recognize the general form of equations for different conic sections
4. Apply knowledge of parabolas to real-world situations

Common Core Standards

• HSG.GPE.2 Derive the equation of a parabola given a focus and directrix.
• HSG.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Prerequisite Skills

Students should be familiar with:

• Basic algebraic concepts
• Graphing on a coordinate plane
• Properties of quadratic functions

Key Vocabulary

• Parabola: A U-shaped curve that is the graph of a quadratic function. It can open upward, downward, or sideways and has a vertex, which is its highest or lowest point. A parabola is also one of the four conic sections formed by the intersection of a plane with a cone.
  Equation:
  y = ax2 + bx + c (vertical parabola)
  x = ay2 + by + c (horizontal parabola)
  y = a(x - h)2 + k (vertex form)
• Circle: A set of points in a plane that are equidistant from a fixed center point. It is a special case of an ellipse where the two focal points coincide.
  Equation:
  (x - h)2 + (y - k)2 = r2
  Where (h, k) is the center and r is the radius.
• Ellipse: An oval-shaped curve formed by the intersection of a plane with a cone when the plane cuts through at an angle. An ellipse has two focal points, and the sum of the distances from any point on the ellipse to these focal points is constant.
  Equation:
  (x - h)2 / a2 + (y - k)2 / b2 = 1 (horizontal major axis)
  (x - h)2 / b2 + (y - k)2 / a2 = 1 (vertical major axis)
  Where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.
• Hyperbola: A pair of mirror-image curves formed when a plane intersects a cone in a way that cuts through both nappes (the upper and lower parts of the cone). A hyperbola has two focal points, and the difference in distances from any point on the hyperbola to these focal points is constant.
  Equation:
  (x - h)2 / a2 - (y - k)2 / b2 = 1 (horizontal transverse axis)
  (y - k)2 / a2 - (x - h)2 / b2 = 1 (vertical transverse axis)
  Where (h, k) is the center, a is the distance from the center to the vertices, and b relates to the distance to the asymptotes.
• Focus: A fixed point used in the definition of conic sections. For a parabola, there is one focus. For an ellipse or hyperbola, there are two foci.
• Directrix: A fixed line used in the definition of a conic section. For a parabola, the distance from any point on the curve to the focus equals the perpendicular distance to the directrix.
• Vertex: The point at which a parabola changes direction. For other conic sections like ellipses or hyperbolas, the vertex refers to the endpoints of the major axis or the closest points to the foci.
• Conic Sections: The curves obtained by slicing a cone with a plane at various angles. The four types of conic sections are parabolas, circles, ellipses, and hyperbolas.

Additional Multimedia Resources

• https://www.media4math.com/MathExamplesCollection--GraphingConicSections

Warm-up Activities (10 minutes)

Identify Conic Sections

• Present students with various cross-sections of a cone
• Ask them to identify and name the resulting shapes (circle, ellipse, parabola, hyperbola)
• Discuss how the angle of the intersecting plane determines the type of conic section

Desmos Activity: Conic Sections Templates

• Direct students to open the Desmos graphing calculator
• Provide them with the following template to graph conic sections: a(x - h)² + b(y - k)² = c, or have students construct it.
• Add sliders for a, b, c, h, and k
• Ask students to manipulate the sliders and observe how changing each parameter affects the shape and position of the conic section
• This template can be used to graph circles or ellipses. 
• To graph hyperbolas and parabolas, different equations are needed. Have students explore what these equations might be.

Real-world Parabolic Shapes Discussion

• Prompt students to brainstorm and share examples of parabolic shapes they've encountered in everyday life
• Possible examples include:
  • The path of a thrown ball
  • The shape of satellite dishes
  • The curve of suspension bridges
  • The cross-section of automobile headlight reflectors
• Briefly discuss how these shapes relate to the properties of parabolas

Teach (25 minutes)

Introduction to Conic Sections

Conic sections are fundamental geometric shapes that emerge from the intersection of a plane with a double-napped cone. These shapes—circles, ellipses, parabolas, and hyperbolas—have deep connections to both geometry and algebra, providing a versatile framework for modeling various real-world phenomena.

1. Geometric Origin of Conic Sections

The term "conic sections" derives from the way these curves are generated: by slicing a cone at different angles relative to its axis.

• A circle results when the plane is perpendicular to the cone's axis.
• An ellipse appears when the plane intersects the cone at an angle but does not pass through the base.
• A parabola is formed when the plane is parallel to the cone's slant height.
• A hyperbola emerges when the plane intersects both nappes of the cone.

These geometric definitions provide a visual foundation for understanding the diverse properties of these curves.

2. Algebraic Considerations

Each conic section has a corresponding algebraic representation that reveals its unique properties. The standard equations for the conic sections are:

• Circle: (x - h)2 + (y - k)2 = r2, where h, k are the coordinates of the center and r is the radius.
• Ellipse: ((x - h)2 / a2) + ((y - k)2 / b2) = 1, where h, k are the coordinates of the center, and a, b are the lengths of the semi-major and semi-minor axes.
• Parabola: y = a(x - h)2 + k (vertical) or x = a(y - k)2 + h (horizontal), where h, k are the coordinates of the vertex.
• Hyperbola: ((x - h)2 / a2) - ((y - k)2 / b2) = 1 (horizontal) or ((y - k)2 / a2) - ((x - h)2 / b2) = 1 (vertical), where h, k are the coordinates of the center.

3. Real-World Applications of Conic Sections

Conic sections are not just mathematical abstractions—they are integral to many practical applications:

• Parabolas are used in the design of satellite dishes and headlights due to their reflective properties.
• Ellipses describe planetary orbits, as explained by Kepler’s laws of motion.
• Hyperbolas are used in navigation systems like GPS and in the analysis of sound waves.
• Circles play a role in engineering, architecture, and even data modeling.

Understanding conic sections equips students to analyze and interpret these real-world contexts.

The next section covers the definition of conic sections, with a focus on parabolas. We'll explore their geometric properties and how to graph them using their standard equations. You can also use this slide show to provide an overview of conic sections:

https://www.media4math.com/library/slideshow/slide-show-conic-sections

Example 1: Circle

Equation in Standard Form: (x - 2)2 + (y + 3)2 = 16

• Center: (2, -3)
• Radius: 4

Graph Description: A circle centered at (2, -3) with a radius of 4. The circle is symmetric about its center.

Example 2: Parabola

Equation in Standard Form: y = 2(x - 1)2 + 3

• Vertex: (1, 3)
• Direction: Opens upward since a = 2 > 0
• Focus: (1, 3 + 1/8) = (1, 3.125)
• Directrix: y = 3 - 1/8 = 2.875

Graph Description: A parabola centered at (1, 3), opening upward with a focus and directrix determining the shape.

Example 3: Ellipse

Equation in Standard Form: (x - 4)2/9 + (y + 2)2/16 = 1

• Center: (4, -2)
• Semi-Major Axis Length: 4 (along the y-axis)
• Semi-Minor Axis Length: 3 (along the x-axis)
• Foci: (4, -2 ± √7) ≈ (4, -2 ± 2.65)

Graph Description: An ellipse centered at (4, -2) with the major axis along the y-direction.

Example 4: Hyperbola

Equation in Standard Form: (x - 3)2/25 - (y + 1)2/9 = 1

• Center: (3, -1)
• Transverse Axis Length: 10
• Conjugate Axis Length: 6
• Foci: (3 ± √34, -1) ≈ (3 ± 5.83, -1)
• Asymptotes: y + 1 = ± (3/5)(x - 3)

Graph Description: A hyperbola centered at (3, -1), opening left and right.

Example 5: Real-World Application (Parabolic Reflector)

Scenario: A parabolic satellite dish is modeled by the equation y = x2/16. The dish's shape ensures that signals bouncing off the surface converge at the focus.

• Vertex: (0, 0)
• Focus: (0, 4)
• Direction: Opens upward
• Dish Dimensions: The parabola represents the cross-section of the satellite dish, with the width and depth determined by specific constraints.

Graph Description: A parabola with its vertex at the origin, focusing all signals or energy at the point (0, 4). The focus represents the position of the receiver in the satellite dish.

To see additional examples of graphs of conic sections, refer to this collection:

Additional Multimedia Resources

• https://www.media4math.com/MathExamplesCollection--GraphingConicSections

Review (10 minutes)

Let's review the key vocabulary terms from our lesson on conic sections:

• Parabola: A U-shaped curve that is the graph of a quadratic function. It can open upward, downward, or sideways and has a vertex, which is its highest or lowest point. A parabola is also one of the four conic sections formed by the intersection of a plane with a cone.
• Circle: A set of points in a plane that are equidistant from a fixed center point. It is a special case of an ellipse where the two focal points coincide.
• Ellipse: An oval-shaped curve formed by the intersection of a plane with a cone when the plane cuts through at an angle. An ellipse has two focal points, and the sum of the distances from any point on the ellipse to these focal points is constant.
• Hyperbola: A pair of mirror-image curves formed when a plane intersects a cone in a way that cuts through both nappes (the upper and lower parts of the cone). A hyperbola has two focal points, and the difference in distances from any point on the hyperbola to these focal points is constant.
• Focus: A fixed point used in the definition of conic sections. For a parabola, there is one focus. For an ellipse or hyperbola, there are two foci.
• Directrix: A fixed line used in the definition of a conic section. For a parabola, the distance from any point on the curve to the focus equals the perpendicular distance to the directrix.
• Vertex: The point at which a parabola changes direction. For other conic sections like ellipses or hyperbolas, the vertex refers to the endpoints of the major axis or the closest points to the foci.
• Conic Sections: The curves obtained by slicing a cone with a plane at various angles. The four types of conic sections are parabolas, circles, ellipses, and hyperbolas.

To further explore these conic sections, you can use the interactive Desmos template below. By default, a circle is graphed, but you can also graph an ellipse, hyperbola, and parabola by adjusting the parameters for a, b, h, and k. Or click on this link to access it.

Activity

Using the Desmos template, graph each of the conic sections by adjusting the parameters a, b, h, and k as provided by your instructor. Observe how changing these parameters affects the shape and position of each conic section.

Quiz

For each equation, identify the type of conic section and provide the requested details (center or vertex coordinates).

1. $\mathit{x}^2 + \mathit{y}^2 = 25$
   
    
2. $\frac{\mathit{x}^2}{16} + \frac{\mathit{y}^2}{9} = 1$
   
    
3. $\frac{(\mathit{x} - 4)^2}{36} + \frac{(\mathit{y} + 2)^2}{25} = 1$
   
    
4. $\frac{\mathit{x}^2}{9} - \frac{\mathit{y}^2}{16} = 1$
   
    
5. $\frac{(\mathit{x} + 3)^2}{49} - \frac{(\mathit{y} - 5)^2}{36} = 1$
   
    
6. $\mathit{y} = \frac{1}{4}(\mathit{x} - 3)^2 - 2$
   
    
7. $(\mathit{y} - 1)^2 = 8 (\mathit{x} + 2)$
   
    
8. $\frac{(\mathit{x} + 5)^2}{64} + \frac{\mathit{y}^2}{36} = 1$
   
    
9. $(\mathit{x} - 2)^2 + (\mathit{y} + 4)^2 = 100$
   
    
10. $\mathit{x}^2 - \mathit{y}^2 = 49$
    
     

Answer Key

1. Type: Circle
   Center: (0, 0)
2. Type: Ellipse
   Center: (0, 0)
3. Type: Ellipse
   Center: (4, -2)
4. Type: Hyperbola
   Center: (0, 0)
5. Type: Hyperbola
   Center: (-3, 5)
6. Type: Parabola
   Vertex: (3, -2)
7. Type: Parabola
   Vertex: (-2, 1)
8. Type: Ellipse
   Center: (-5, 0)
9. Type: Circle
   Center: (2, -4)
10. Type: Hyperbola
    Center: (0, 0)