The first thing I did when I was given this project, was to pull out my workbook and past graphing worksheets so I could get a glimpse at the functions I would be using and what they would look like. For example, body parts that were round such as the head, eyes, and wings would most likely use an exponential or semi-circle function. I created a rough sketch of what I wanted my holiday card to look like and then went in a labeled the drawing with the functions I thought could be used to achieve the shape. For example, I picked a sum of a linear and sine function for the bottom of the wings because the sine function would allow for the bottom of the wings to by wavy and I knew that when a linear function is added, the sine function would become slanted. For the nose I knew would need a line that would have two bumps in it. I remembered the Factoring Polynomials lesson and how I could manipulate how a line curves with the factors and the degree of each factor. As a result I created the quartic function : (x^4) – (0.82x^2) – 0.5519.
One of the challenges I faced was created the letters, in particular the slanted lines of the letters. I played around with the slant until I was happy with how it looked in comparison to how tall the letters would be. In order to find the correct y-intercept for the linear equation so that it would be in the correct spot, I would find key points the lines would have to pass through then using those (x,y) coordinates I could find the b value.
Restrictions of the domain and range greatly helped me achieve my desired shapes for this graph. In order to make sure that different lines would perfect line up together, I’d use the points of intersection to determine which values to use. For example, I used the point of intersection between the exponential function of the wing and the exponential function of the body to find the minimum and maximum value of x for the respective functions.
This assignment greatly helped me develop a better understanding of transformation of functions and relations. For each section, I started off with one function and then transformed it in various ways to complete the section. For example, the eyes and eyeshadow were created by transforming the same quadratic function in different ways. In this case, I translated it up and down as well as reflected it along the y-axis. I also grew quite comfortable with stretches. Most, if not all, of the functions used in the card have been stretched. For example, I didn’t want my head to be a perfect circle as that’s not what my head is like in real life. Therefore, I used stretches to manipulate the semi-circle function. The top part of the head has been vertically stretched by a factor of 1.3 while the bottom was vertically compressed by a factor of 0.9. This assignment also helped become better at visualizing what these functions look like and remembering how the different transformation affect functions.
