Conics – Final Project Option
A LGEBRA II WITH T RIGONOMETRY
Name ________________________
Date ____________ Pd _________
In this project, you will investigate the four types of conics, two of which you have seen
before. A conic can be defined in two ways. First, it is any cross-section of a cone. For
example, a flat (horizontal) cross section yields a circle. Second, it is any equation
where either the x or the y is raised to the second power. For example, the equation of
the unit circle is x 2  y 2  1 .
I.
Make four cones. A good way to make a cone is to trace a large circle, then
cut it in half so you have two semicircles. For each one, tape the straight
diameter to itself (one half to the other half, that is) and you have a cone!
II.
Circles
2
2
1) The general equation of a circle is x  h    y  k   r 2 . There are two
variables: x and y, and there are three parameters: h, k, and r.
2) Find out what the h, k, and r represent in the graph of a circle by
researching circles and the general equation of circles. An excellent
resource for this is the online encyclopedia www.wikipedia.org . Another
excellent resource is Chapter 10 of your textbook.
3) Roll a die for each parameter above, and write your personal circle
equation here:
4) Solve your equation for y (you will have two equations, positive and
negative). Check with Mr. Yates that you have done 3 & 4 correctly.
5) Enter your equation in your calculator or graphing software (for your
calculator, please ZoomSquare to get a more accurate picture than
ZoomStandard offers). Print or sketch your graph.
6) Highlight important points or parts of your graph (like the circle’s center
and radius). Also, give coordinates of two other points on your graph.
7) Cut a flat (horizontal) cross-section of one of your cones. The outline
should be a circle. Trace it on your paper. You should also turn in the
dissected cone, with the label “Circle” written on it.
III.
Ellipses
1) Research the general equation of an ellipse. Identify the variables and
parameters.
2) Roll a die for each parameter, and write your personal ellipse equation
here:
3) Solve your equation for y (you will have two equations, positive and
negative). Check with Mr. Yates that you have done #1 – 3 correctly.
4) Enter your equation in your calculator or graphing software. Print or
sketch your graph.
5) Highlight important points or parts of your graph (like the ellipse’s center,
major and minor axes, extra credit for its foci). Also, give coordinates of
two other points on your graph.
6) Cut a diagonal cross-section of one of your cones. The outline should be
an ellipse. Trace it on your paper. You should also turn in the dissected
cone, with the label “Ellipse” written on it.
IV.
Parabolas
1) Research/Recall the general equation of a parabola. This should look
familiar!!! Identify the variables and parameters.
2) Roll a die for each parameter, and write your personal parabola equation
here:
3) Solve your equation for y (you will have only one equation this time).
Check with Mr. Yates that you have done #1 – 3 correctly.
4) Enter your equation in your calculator or graphing software. Print or
sketch your graph.
5) Highlight important points or parts of your graph (like the parabola’s
vertex, axis of symmetry). Also, give coordinates of two other points on
your graph.
6) Cut a cross-section parallel to one side of one of your cones. The outline
should be a parabola. Trace it on your paper. You should also turn in the
dissected cone, with the label “Parabola” written on it.
V.
Hyperbolas
1) Research the general equation of a hyperbola. Identify the variables and
parameters.
2) Roll a die for each parameter, and write your personal hyperbola equation
here:
3) Solve your equation for y (you will have two equations, positive and
negative). Check with Mr. Yates that you have done #1 – 3 correctly.
4) Enter your equation in your calculator or graphing software. Print or
sketch your graph.
5) Highlight important points or parts of your graph (like the hyperbola’s
two vertices, central rectangle, and asymptotes). Also, give coordinates of
two other points on your graph.
6) Cut a vertical cross-section of one of your cones. The outline should be
half a hyperbola. Trace it on your paper. You should also turn in the
dissected cone, with the label “Hyperbola” written on it.
Rubric:
Part I –
4pts (1pt per cone made)
Parts II–V – 24pts each, as follows
4pts for research / identifying the parameters
4pts for writing the equation, generated with random values 1-6
4pts for solving for y
4pts for accurate sketch or print of graph
4pts for important points highlighted
4pts for labeled cross section of cone
Total: 100pts