Jack draws a Penguin & asks Don to help him describe it using the graphs of equations!

It took us about 4 months to finish, but we did it. The # of the equation is circled.

Each equation led to a lot of discussion- multiplying signed numbers in 1),  the equation of a circle in 4), slope and intercept for the lines 2), 3), 5), and 6),... and what the limits of the x and y values needed to be in all of them.

For the top part we tried the parabola x² = y, then ^-x²=y to open downward, but that was too thin. We ended with a wider parabola ^-(1/2)*x² =y (circled 1) with ^-4<=x<=4 to approximate Jack's picture. Note ^-8<=y reads y is greater than or equal to ^-8.

The under-flap equations 2) and 3) are both linear with opposite slopes: 2). is (1/2)*x-4=y with x-limits  ^-4<=x<=^-2  and 3). ^-(1/2)*x-4=y with x-limits 2<=x<=4, this 4 corrects the 3 on the graph.

For the body we decided on parts of a circle. and discussed the equation of a circle (left, near the bottom). We came up with equation 4).  x² + (y+6)² = 2.5²   a circle with radius 2.5, and the center at (0,^-6) and ^-8<=y<=^-5.

For the outside of the legs we used linear graphs with opposite slopes: 5). y=^-1x-7 and 6). y=1x-7, both with the y-limits are the same: ^-9<= y <=^-7 and ^-9<= y <=^-7. For the inside of the legs 7).we used the absolute value function |x| = y with the x-limits ^-1<=x<=1. N.B. |3|=3 and |^-3|=3.

The bottom of the feet we came up with 8). as  y= ^-10 with the x-limits ^-2<=x<=2.

Great job, Jack!!!