Section 12.2 Implicitly Defined Curves and Cicles

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

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

Section 12.3 Ellipses

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?

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

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

Section 12.4 Hyperbolas

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

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

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

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