1. Section 12.2 Implicitly Defined Curves and Cicles

2. Recall that the equation for a circle centered at (h, k) with a radius of r is
This is know as an implicitly defined curve because we cannot explicitly solve the equation for x or y
Consider
If we tried to solve the equation for y we would get

3. What if we wanted to plot y2 = x2?
What do we have with the equation

4. Section 12.3 Ellipses

5. Let’s take a look at what is going on with the following equation.
Notice it is still centered at
(0, 0)
It is narrower along the
x-axis
What are the horizontal and
vertical axes?

6. Let’s look at the same ellipse with a different center
This equation for this ellipse is
The major axis is 2 and the
minor axis is 4

7. With a little exploration from the previous examples we see that the general equation of an ellipse centered at (h, k) with a horizontal axis of 2a and a vertical axis of 2b is
Let’s find the center and lengths of the major and minor axes of the following ellipse

8. Section 12.4 Hyperbolas

9. Now we know the following gives us an ellipse
But what would happen if we replace the + with a -?
Let’s take a look at the graph

11. These form a hyperbola that is centered at (1, -2), is stretched vertically by a factor of 2, and is stretched vertically by a factor of 4

12. What do you think happens if we reverse the pieces of the equation

13. The implicit equation for a hyperbola that opens to the left and right is given by is given by
Its asymptotes are diagonal lines that through the corners of a rectangle of width 2a and height 2b centered at the point (h, k).
The graph of
is similar, but opens up and down