The document discusses the lemniscate curve defined by the equation (x^2 + y^2)^2 = 2a^2(x^2 - y^2). It provides steps to graph this curve using a graphing calculator and describes how the parameter a affects the shape of the curve. It also determines the points where the tangent line is horizontal and discusses potential real-world applications of the lemniscate curve, such as racetracks and figure skating.

1. Eight Curve Annie Tan Alice Hu Junwei Kwee Johnny Chang

2. Lemniscates The lemniscate is official term describing the eight curve. The equation of a lemniscate is x 4 =a 2 (x 2 -y 2 )

3. The Problem: P.161 #7 The graph of the eight curve, (x 2 +y 2 ) 2 =2a 2 (x 2 -y 2 ), is shown. A. explain how you could use a graphing utility to graph this curve B. Use the graphing utility to graph the curve for various values of the constant a. Describe how a affects the shape of the curve C. Determine the points on the curve where the tangent line is horizontal. -a a

4. Solving (a) A. explain how you could use a graphing utility to graph this curve x 4 =a 2 (x 2 -y 2 ) X 4 =a 2 x 2 -a 2 y 2 A 2 y 2 =a 2 x 2 -x 4 Y 2 =(a 2 x 2 -x 4 )/a 2 Y= +/- ((a 2 x 2 -x 4 )/a 2 ) 1/2 = +/- ((a 2 x 2 -x 4 ) 1/2 )/a Plug + ((a 2 x 2 -x 4 )/a 2 ) 1/2 and - ((a 2 x 2 -x 4 )/a 2 ) 1/2 into the calculator. Substitute values for a because a is a constant.

5. Solving (b) B. Use the graphing utility to graph the curve for various values of the constant a. Describe how a affects the shape of the curve

6. Solving (b) cont. As we can see from the previous graphs of the curve, as a increases, the curve will get taller and thinner and the y-values of relative extrema will increase.

7. Solving (c) C. Determine the points on the curve where the tangent line is horizontal. x 4 =a 2 (x 2 -y 2 ) 4x 3 =a 2 (2x-2yy’) 4x 3 =2a 2 x-2a 2 yy’ 2x 3 =a 2 x-a 2 yy’ Y’=(a 2 x-2x 3 )/a 2 y 0=(a 2 x-2x 3 )/a 2 y a 2 x-2x 3 =0 a 2 =2x 2 X=+/- a/(2) 1/2

8. Solving (c) cont. X 4 =a 2 (x 2 -y 2 ) a 4 /4=a 2 (a 2 /2)-a 2 y 2 a 2 y 2 =a 4 /2 – a 4 /4 a 2 y 2 = a 4 /4 y 2 = a 2 /4 y= +/- a/2 Points: (a/(2) 1/2 , a/2), (a/(2) 1/2 , -a/2), (-a/(2) 1/2 , a/2), (-a/(2) 1/2 , -a/2)

9. Real World Application Racetracks We can also use this curve for a car racing track. For the racers, the relative extrema are important points because that is where the motion of the cars change the most. Y 2 =(a 2 x 2 -x 4 )/a 2 2yy’=(2a 2 x-4x 3 )

10. Real World Application Ice Skating http://www.youtube.com/watch?v=Y3jsq-u8pWM A figure-eight in ice skating resembles the shape of an eight curve.

11. Facebook Integration Alice: http://www.facebook.com/#!/aliice?ref=ts Annie: http://www.facebook.com/#!/annietanniebannie?ref=ts Johnny: http://www.facebook.com/#!/AzNkiSEkiStaR Junwei: http://www.facebook.com/#!/profile.php?id=100001146442174&ref=ts

12. Importance of Facebook It’s scientifically proven that people cannot live without social interaction. Websites like Facebook and Myspace help people interact with people Companies view Facebook profiles during application and interview processes. Basically, what one displays on their Facebook page is what the interviewers and managers see.