Study Guide > Cleverly Concealed Conics Portfolio (1) - Texas Connections Academy_ Houston PRE CALC

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Cleverly Concealed Conics Portfolio Directions: ​For this portfolio, you will find examples of the four conic sections in the world around you. Once you have examples of each conic section, you will choose one to analyze more closely. You will generate the equation and find the important parts of the conic section.

Then you will move the foci and write a paragraph explaining the implications of the move for the application. Finally, you will reflect on your conic exploration. Be sure to complete all four parts of the portfolio before submitting it to your teacher.

Part 1: Conic Examples Find real-world examples of each conic section. You may use items or structures in your surroundings, or may search for items on the Internet.

Complete the tables for each of your models. Provide as much detail as possible to explain how these are good examples of the conic sections. Be sure to cite your source and provide a photo or a sketch.

Record your notes for your example of a circular structure in the following table.

Conic: Circle

Notes

Example

Ferris wheel

Source

Internet - London Eye

Photo

Description of Cleverly Concealed Conic

This cleverly concealed conic is a perfect example of a circle being put to use in the real world. It is a world famous attraction that was beautifully built and though it is quite obvious that it is a circle, its conic properties are more concealed.

Conic: Ellipse

Notes

Example

Football

Source

Internet

Photo

Description of Cleverly Concealed Conic

This cleverly concealed conic is a perfect example of an ellipse being put to use in the famous sport that is American football. The shape of the football, that of an ellipse, is what defines the sport and what separates this ball from all other ones. Its shape gives it unique aerodynamic properties and surprisingly, given the sport’s popularity, the fact that it is an ellipse is largely unknown.

Conic: Parabola

Notes

Example

Banana

Source

Internet

Photo

Description of Cleverly Concealed Conic

This cleverly concealed conic is extremely concealed in my opinion and a perfect example. This is also a naturally occurring conic, whereas the other three conics I have mentioned are man-made. This just goes to prove the connection between mathematics and the natural world and it is very interesting to see.

Conic: Hyperbola

Notes

Example

Hourglass

Source

Internet - Zulily

Photo

Description of Cleverly Concealed Conic

This cleverly concealed conic is a perfect example of a hyperbola and it can be very concealed to someone who is not too familiar with conics. It is also interesting how this shape is simply referred to as “hourglass” and never hyperbola; not many people realize its conic properties.

Part 2: Conic Graph and Equation Choose one of your conic examples to study in greater detail. Complete the table to create the graph and equation for your chosen example.

1. If possible, use a photo of the structure or item. Otherwise, draw a detailed sketch.

2. Measure and label the key dimensions of the shape in order to duplicate them in a to-scale graph. Be sure to label your units of measure to reflect the actual measurements.

3. Draw the underlying conic section on a coordinate grid.

4. Identify the coordinates of the appropriate key parts of the conic section.

5. Determine the closest approximation of the equation in standard form for the conic section.

Notes Conic Example

Circle

Photo or Sketch This part can be done in a standalone file.

Measurements Include the key parts of the chosen conic section.

Diameter of 120 metres Radius of 60 meters - Source: Britannica.com

Graph of Conic Section You may use separate grid paper or copy and paste a screenshot or photo of your graph.

Equation

(x – h^)2 + (y – k)^2 = r^2

Use standard form for the conic section you chose.

With origin as center: ​x^2 + y^2 = 3600 I’m going to use the x-axis as the “ground” to make my graph somewhat more realistic.

The new equation is: ​x^2 + (y-60)^2 = 3600

Part 3: A New Graph Double or half the distance from the foci/focus to the vertices/vertex, and determine the new key points on the graph. Calculate what happens to the equation.

Compare the new graph to the original graph. Write a paragraph explaining what would happen if your original example were changed in that way. What impact would the shape change have on the function served by the example?

New Conic Section

Notes

Conic Example Double or half

Distance from focus (center) to vertex = radius = 60 Half of 60 = 30

New Key Points

Center/focus: (0,30) y-intercepts: (0,0) and (0,60)

New Equation

x^2 + (y-30)^2 = 900

Graph of Revised Conic You may use separate grid paper or copy and paste a screenshot or photo of your graph.

Impact on Real-World Making the London Eye half its size would not impact its functionality.

Use or Purpose

However, the London Eye is such a big tourist attraction partially because of its astounding size. If halved in size, the London Eye may lose tourists.

Part 4: Reflection Write a paragraph explaining what you learned about conics and describe any challenges you encountered.

I learned that conics are much more abundant in everyday life than I previously thought. I was aware that circles would be very common however, I wasn’t expecting to see parabolas and hyperbolas in such common objects, like bananas and hourglasses. A challenge I encountered was when I was writing my equations, I struggled writing an equation given only a radius, so I started off writing an equation at the origin and moving on from there.