M2 PreCalc Investigation – Sections 6.2-6.4 – Intro to Conic Sections
In this investigation, you will learn how to use Desmos to graph equations, identify types of conic
sections from a graph or equation, and identify key parts of circles, ellipses, and hyperbolas from the
graph or the equation.
Part 1: Introduction to Desmos
1. Choose either working on a laptop or on a mobile phone.
a. You can either head to www.desmos.com
b. Or you can download the app from the AppStore
2. In this investigation, we will be graphing equations that are not in “y=” form. “y=” equations are
called “explicit.” The equations we will graph today are written in “implicit” form.
3. Select start graphing!
4. Use the keyboard to type in: x 2  y 2  16
5. What shape is the graph of x 2  y 2  16 ?
6. What are the center and length of the radius for this circle?
7. How is it different if we graph:  x  2    y  3  16
2
2
Part 2: The Ellipse
8. Now we’re going to look at a different shape. In the input bar type in
x2 y 2

 1 . You can turn off
9 16
the first equation by:
clicking this button.
9. This shape is called an ellipse. Where have you heard the word “elliptical” before?
10. Think of an ellipse as a stretched-out circle. Ellipses and circles both have a center, but an ellipse has
many different diameters or radii. The longest diameter is called the major axis, and the shortest
diameter is called the minor axis. Where is the center of this ellipse? What are the lengths of its
major and minor axes? What are the lengths of the longest and shortest radii?
11. The vertices of an ellipse are the endpoints of its major axis. What are the vertices of this ellipse?
x2 y2

 1 , and graph. (You can do this by
12. Switch the denominators in the equation so that it reads
16 9
clicking on the equation in the left or type in a new equation in input). How did the graph change?
Find the center, vertices, major/minor axes, and longest/shortest radii.
Part 3: The Hyperbola (pronunciation tip: emphasize the second syllable)
x2 y 2
13. In Desmos, change the “+” to a “- “ in the equation and graph. (Or Input:

 1)
16 9
14. The graph of this new equation forms a hyperbola. Like circles and ellipses, hyperbolas have a
center. Like a parabola, hyperbolas have a vertex on each section of the graph; so it has two
vertices. Where are the center and vertices for this hyperbola?
y2 x2
15. Reverse the order of subtraction in the equation so that it reads
  1 , and graph. How did
9 16
the graph change? Where are the center and vertices?
Part 4: Comparing Equations of Circles, Ellipses, and Hyperbolas
16. What do you notice about the similarities and differences between the equations of circles, ellipses,
and hyperbolas?
17. Write the equation of a circle, ellipse, and hyperbola. Graph each, and identify the key parts of each
shape. (Hint: This will be a lot easier if the constants in your equations are perfect squares.)