So here are the solutions to quiz 2 from Unit 5.2 The first questioned asked about r equals 1.8 What value is the fixed point? So, I will zoom in here. It looks like this region has 1.8 in it. Zoom in a little bit. And now I'm going to move the cursor Until the r value is about 1.8 Then I will see what the x value is. So, here where my cursor is r is a little more than 1.8 and I see that the x value there is 0.436 So about 0.44 So the best answer to this question was 0.44 There is a fixed point at around .44 when r is 1.8 The next question asked about bifurcation from period two to period four. So I'm going to go back and I'm going to zoom in here. So bifurcation from period 2 to period 4 that occurs around here. So when there are two lines if you go up from here and the line crosses this imaginery line intersects two lines in the bifurcation diagram that means it would be a final state diagram with two dots Here we go up we cross through four so it is period four. The transition occurs here Here. And that looks to be about 3.44, 3.45. So that gives us an approximate r value where this bifurcation occurs. The last question, question 3. Asked what goes on around 3.906. There is a periodic window of what period So let me go back one. And this will take a little bit of trial and error exploration. So let me zoom in. 3.9 where is that. That looks to be around here. I want 3.906. Let's try zooming in. Here. Update again. 3.906. 3.906 Here is around 3.906 You can tell because I'm reading this mouse position here. Let's see 1,2,3,4,5. So I see period five behavior here. So this is a window of period five. So, near r equals 3.906. There is a periodic window of period five.