So Logan has devised a logo for her company which is a square divided into 3 sections with 2 functions. Our objective is to find what functions fit the two curves on her logo and also make it fit on two other sizes of her logo. Just looking at the shape of the lines 2 types of functions immediately jump into my head, sinusoidal and cubic. I first traced a grid onto the original copy of the logo I was given in order to get points to start trying to form a function to match the design. The grid I used had 4 squares per inch. I then plotted 7 points on each curve shown in the table below. Upper Curve| Lower Curve|
X| Y| X| Y|
0| 4| 0| 1|
1| 2| 2| 1|
3| 2| 5| 4|
5| 5| 6| 5|
7| 8.5| 8| 6|
9| 9.5| 9.5| 5|
10| 9| 11| 3|
I then plotted the points for the upper curve on my calculator so I could adjust a function and see how it compared to the original points instantly. I started out with the basic sin function which was of course way off.
I knew I had to increase the amplitude to get it to reach the highest point and I estimated that the midline of the sinusoid would be at about 5 so that would mean I would have to add a vertical shift as well. I changed the function to be 4sin(x)+5 which was still off.
I saw that I had to change the period because it was not rising and falling with the points as well as flip the amplitude because it was high while the points were low and vice versa. I came up with -4sin(0.5x)+5 I made a few more adjustments because the amplitude I had picked (4) was too high for the highest point (9,9.5) and because the function was rising and falling too quickly so I had to shrink the period. I also had to add a horizontal shift to the left to better make the function fit. I did this by adding something to the x within the parentheses. After all of those adjustments I came up with -4.3sin(0.55(x+0.9))+5 shown below
I then decided to try a cubic function because I thought it...