IB Mathematics SL IA
The Use of Functions and Relations to Draw a Car
Suleman Tariq
In this assignment, I use the online graphing calculator software Desmos, and Ti-84 Plus to construct and graph the equations used in this assignment.
Candidate Declaration: I confirm that this work is my own and is the final version. I have acknowledged each use of the words, or ideas of another person, whether written, oral or visual
Candidate’s Signature: _________________________ Date: ___________________________
Background:
The history of cars goes way back to the 1800s. The first car was invented by Karl Benz in 1886 (Daimler, 2018). The first car was driven in Mannheim, Germany, reaching a top speed of 16 km/h (10 mph). As time passed, the variety in the cars increased and the cars became more efficient.
I have always been into cars and the way they work as a child and still am to this day. As my curiosity increased, I started to research about cars and learned about the different parts and their function. It had always fascinated me that we were able to include so many things in a small place and how each part of a car cooperates with various other parts to perform its function.
In Pakistan, my dad owned a car dealership which was passed down to my dad by my grandfather. This exploration is for my grandfather who passed away in 2006 and I am doing this in his memory. I have a dream that I will continue this legacy and make my grandfather proud of what I have accomplished.
My dad would bring me with him every weekend to the shop. I have seen the car shop develop and become an established business. Every year, I would see new cars arrive and would experience the excitement in people to see the new cars.
I wanted to somehow incorporate the topic of cars in this exploration. In math, it had always fascinated me that each of the functions has a different shape and curvature. I had found myself drawing simple smiley faces using mathematical functions and relations for fun on desmos.com. I loved the artistic side of math and how by using simple restrictions you were able to create something extraordinary. After putting thought into how I was going to incorporate the topic of cars into Math, I realized that I was able to draw a simple car using functions and relations. This topic allowed me to include both of my interest.
Aim of this IA
The aim of this Math IA is to draw a car using functions and relations. I wanted to create an art piece by just using math. This shows that the use of math can also be used in something beautiful like art. I will be using Desmos in order to make my car. Throughout the exploration, I will be explaining why each of the functions was used. This picture I will be using as a template is shown below.
(Sweet Clip Art, 2018)
Equations:
In order to graph this car mathematically, you cannot use just one equation. This is because there is no function that has a shape of a car. I must use various equations with restrictions applied to them. Restrictions limit the domain and the range of a function and restrictions can be added by including curly brackets (also called braces): {}. This is significant as restrictions allow me to use parts of the functions in order to create the car. I will also need to include the horizontal and vertical compression as that allows me to create variations in functions. Moreover, vertical and horizontal translations will also be applied to the functions so that they fit exactly where they need to go. First of all, the car has to be broken down into different functions.
Equation 1:
I did the body of the car fist as by knowing the structure of the body, it made it easier for me to make the windows and the wheels. This expresses the top part of the car. I wondered if I should use a parabola or an ellipse for the body of the car. Both have a round shape which is what I needed. First, I needed to understand parabolas and ellipse before making the decision.
Quadratic:
I use quadratic functions throughout this exploration, for example to make the windows as well as some other details. The parent function of the quadratic formula is y=x2. The function could also be expressed in its vertex form which is: . The a value of the parent function is 1 while c and d are both 0. In order to fit my needs, I alter the a, b, c and d values in order to vertically stretch, reflect the function over the x-axis, to translate the function horizontally or vertically. (c,d) represents the vertex of the parabola. For example, the vertex of the parent function is (0,0) as the c and d values are 0.
a value
b value
c value
d value
(a<0):
The function gets reflected over the x-axis.
Ex. is reflected across the x-axis
(b<0):
The function gets reflected over the y-axis.
Ex. is reflected across the y-axis
(c<0):
The function gets shifted horizontally to the left.
Ex. is horizontal translation to the left by 3 from the origin, (0,0).
(d<0):
The function gets vertically shifted down.
Ex. is vertical translation down by 3 from the origin, (0,0).
(0<a<1):
The function gets vertically compressed.
Ex. is vertically compressed by a factor of 0.5.
(0<a<1):
The function gets horizontally stretched.
Ex. is horizontally stretched by a factor of 2.
(c>0):
The function gets shifted horizontally to the right.
Ex. is horizontal translation to the right by 4 from the origin.
(d>0):
The function gets vertically shifted up.
Ex. is vertical translation up by 4 from the origin.
(a>1):
The function gets vertically stretched.
Ex. is vertically stretched by a factor of 3
(a>1):
The function gets horizontally compressed.
Ex. is horizontally compressed by a factor of
Ellipse:
I use ellipses and circles in this exploration because of their shape. The car that I have chosen to draw contains circles and ellipses. An ellipse contains a major axis and a minor axis. The major and minor axes of an ellipse are diameters of the ellipse. The major axis is the longer diameter while the minor axis is the shorter diameter.
The general function of a horizontal major ellipse is:
The general function of a vertical major ellipse is:
In the general equation, the a value is half the length of the major axis and the b value is half the length of the minor axis (a>b). In a horizontal major ellipse, the major axis is in the horizontal axis (y-axis) and in a vertical major ellipse, the major axis is in the vertical axis (x-axis). When the larger number is underneath the x, then the ellipse is horizontal major ellipse and when it is under the y, then it is vertical major ellipse.
I started off using a quadratic function and tried to stretch the function horizontally by making the b value small to make the top part of the car, however I was unsuccessful. I realized that parabolas come to a point (vertex), which made it almost impossible to create the dome shape I was looking for.
After the unsuccessful try, I tried to use ellipses. I wondered if I needed a horizontal major ellipse or a vertical major ellipse. In this case I needed a horizontal major ellipse to make that dome shape of the car. Now I wanted to find the specific values that gave me the most desired shape.
It was observed that having the b2 value at 4 and the a2 value at around 9.5 gave me the perfect shape that I wanted. Since I wanted a horizontal major ellipse the a2 value needed to be larger than the b2 value. However, if a2 value was lot larger than the b2 value, then the top part would be too flat.
General Equation:
When that was decided, I had to apply the horizontal and vertical translations to the function. Since this was the first function I had graphed, a vertical and a horizontal transition was not needed, and my other functions could have a vertical and horizontal translation and could revolve around this function. However, I wanted the middle of the car to be around the origin (0,0). Using that knowledge, I horizontally translated the function 0.6 units to the right. The h value is responsible for the horizontal translation so 0.6 can be seen where the h value is. The function was also vertically translated the function 0.65 units up and this can be seen where the k value is (the k value is responsible for vertical translations).
Now all that was left was the restrictions. The values of the domain restriction were based on intersection points of the functions surrounding equation 1. When I made equation 2, the intersection between Equation 1 with equation 2 was used. Since an ellipse does not pass the vertical line test as it has 2 y-points associated with every x-point. I had to set a restriction on the range as well. Using the y-values of the 2 intersections coordinates and I was able to set a range restriction as well to ensure that each of the x-point had only one y point associated with it.
Equation 2:
This horizontal parabola has gone through the following transformations:
• The function is reflected across the y-axis
• The function has been horizontally shifted 4.4 units to the right
• The function has been vertically translated 0.38 units down.
• The function has been horizontally compressed by a factor of 0.28
A horizontal parabola is a parabola that opens to the left or right. The general equation of a horizontal parabola is:. The a value is responsible for horizontal stretches and compressions. The b value is responsible for vertical stretches and compressions. The c value shifts the graph vertically while the d value shifts the graph horizontally. These transformations are different from the vertical parabola. The reason the b value and the c value are now responsible for vertical transformations instead of horizontal transformations is because they are attached to the y part of the function.
I wondered what kind of function would transition nicely with top part of the car (which was an ellipse). I tried an ellipse and a circle, but I was unsuccessful with both. I needed something like a parabola. As I mentioned before parabola have a vertex and I wanted to use the vertex part of the function. A standard parabola was not going to work because it opens up or down. This is why I decided to use a horizontal parabola. The value of a was negative as I needed the graph to open horizontally towards the left to imitate the back portion of the car. Using the a value of 0.28 gave the back portion of the car a gradual a gradual curve. I found that using 0.2 as the a value made the function too wide while 0.4 as the a value made the function too narrow so 0.28 was perfect.
A horizontal translation of 4.4 units to the right and a vertical translation of 0.38 down worked well as the back of the car perfectly sloped down and the function was in sync with equation 1. The horizontal translation represents the d value while the vertical translation represents the c value in the general equation.
A range restriction was set using the intersection points between the backlights and the top part of the car. I would use the y point of the intersection points in order to find out range because a horizontal parabola is not a function (does not pass the vertical line test) so in this case a domain restriction would not be useful. I found out that y coordinate of the intersection coordinate between equation 1 and equation 2 is 1.58. It was later found out that the y point of the intersection between backlight (equation 10) was -0.41. Using these intersection points I was able to this restriction on the range: .
The red functions represent the new functions while the black functions represent the previous functions.
Equation 3,4,5,6:
Front wheel:
Equation 3 (small circle)
Equation 4 (big circle)
Transformation:
• Horizontally translated 2.65 units to the left
• Vertically translated 1.45 units down
The circles have the centers at (-2.65, -1.45)
Back wheel:
Equation 5 (small circle)
Equation 6 (big circle)
Transformation:
• Horizontally translated 2.7 units to the right
• Vertically translated 1.45 units down
The circles have the centers at (2.7, -1.45)
In this exploration, I also use a lot of circles, for example the wheels of the car. Circles are a variation of ellipses. As mentioned previously, ellipses have a major and minor axis. If the major and minor axis are equal, the ellipse is a circle. The general equation of a circle is:. The center being at the point (h, k) and the radius being r.
The radius or the r value is obtained by square rooting the r2 value. For example, in equation 3 and 5, the r2 value is 0.3 therefore radius=, which is about 0.55. Equation 4 and 6 have a radius of which is about 1.10.
It would make sense to use a circle in order make the wheels of the car. Each wheel has an inside circle which represents the rims of the car. The inside circle (rims) of the wheel is the same when compared to the big circle, and the only thing that is different is the radius of the 2 circles. This ensures that the small circle is right in the middle of the big circle because both of their centers are the same. I wanted to have the front when and the back wheel to have the same size. I did not want the radius to be too big or too small, have the radius at 1.1 made the most sense as the ratio of body to wheels looked perfect.
Furthermore, the circles were also horizontally and vertically translated. Both of the wheels have been vertically translated 1.45 units down to ensure that both of the wheels are on the same level and this can be seen where the k value is in the general equation which is responsible vertical translating the circles. The h value in the general equation is responsible for the horizontal translation.
When I was doing this part of the car, it reminded me of the time when me and my grandpa tried to build a car in the shop in Pakistan. He explained to me the different parts of the car and how they work. That’s when I got really interested in cars. One thing I remember asking my grandpa was, “why are the wheels round and not any other shape?” My grandpa then built a small car with square wheels and asked me to push it, and then he did the same with circular wheels. I remember that the car with circular wheels was easier to move. After the demonstration, he explained to that if the wheel car had been any other shape, it would be difficult to move the car.
I did not put a restriction on the circles because all the circle was needed.
Equation 7 and 8:
Equation 7:
Equation 8:
Demos won’t let me graph functions with multiple restrictions, so I had to copy and paste the functions with different restrictions. These 2 equations are just horizontal lines. I used y=-1.8 and y = -1.12 as that’s where the horizontal lines were needed on the car. There were a lot of intersection points as it ran through both wheels and cam in contact with the headlights and backlights
Based on the intersection points, I was able to set the restrictions on the horizontal lines.
Equation 9 and 10
Equation 9 (headlights)
• The circle has been horizontally translated 4.55 units to the left and vertically translated 0.75 units down, and has the center at (-4.55, -0.75)
• The circle has the radius of about 0.37
Equation 10 (backlights)
• The circle has been horizontally translated 4.55 units to the right and vertically translated 0.75 units down, and has the center at (4.55, -0.75)
• The circle has the radius of about 0.37
I wonder if I should use a parabola or a circle. I decided to use a circle as the headlights, because of the curvature of the circle. My car has very smooth edges and because of that, a circle would be a good choice. A parabola would not work in this case, because I would not able to get the shape I desired because a circle has curves uniformly while a parabola does not.
As previously mentioned, I wanted to have the centre of the car at the origin (0,0), and as a result I was able to reflect the one of the circles across the y axis.
In this case the r2 value is 0.14 therefore the is radius= is about 0.37. The headlights of the car are pretty small, which is why I didn’t want the radius to be too big.
For equation 9 and 10, intersection coordinates between the surrounding equations were used to restrict the domain and the range.
Equation 11 and 12:
Equation 11 (front bumper):
Transformations:
• The function has been horizontally stretched by a factor of 2
• The function has been vertically shifted 1.45 units down.
• The function has been horizontally shifted 4.9 units to the left.
This function has the vertex at (-4.9,-1.45)
Equation 12 (back bumper):
Transformations:
• The function has been horizontally stretched by a factor of 2
• The function has been vertically shifted 1.45 units down.
• The function has been horizontally shifted 4.9 units to the right.
• The function has been reflected across the y axis.
This function has the vertex at (4.9,-1.45)
Horizontal parabolas were used to make the front and back bumper of the car. Both of the functions are very similar however there were some differences in the transformations of each of the functions. I thought about using circles for the bumpers of the car but then it made bumpers would look a lot like the headlights and the backlights.
Both of the equations have a vertically stretched by a factor of 2. I found out that by using a horizontal stretch by a factor of 2 would give me the best shape of a bumper. I tried a horizontal compression by a factor of 0.5, but then again, the bumpers of the car would look a lot like the headlights and the backlights of the car. If I used a horizontal stretch by a factor of 3 or higher, the function would get too stretched and would make the bumper look to sharp. Equation 8 had to be reflected across the y axis as I wanted the parabola to open to the left for the back bumper.
I used transformations on the functions so that they are located where they are supposed to be. I used the intersection coordinates to restrict the range of the functions.
Equation 13:
I wonder if a circle is the best option when it comes to making the hood of the car. I tried using a circle, but I could not get it to look like a hood as it looked too oversized. I tried using a horizontal major ellipse like I used an ellipse to make the top part of the car (equation 1). However, I was again unsuccessful. I needed an ellipse that was tall and skinny, so I decided to use a vertical major ellipse instead of a horizontal major ellipse.
The center of the function was horizontally translated 3.3 units to the left and vertically translated 1 unit down to sync with the other functions. In order to set the domain and the range restriction intersection coordinate between equation 13 and equation 6 and the intersection coordinate between equation 13 and 15 were used.
Equation 14
Transformations:
• Vertically compressed by a factor of 0.9
• Horizontally translated 2.9 units to the left
• Vertically translated 0.75 units up
This parabola has a vertex at (-0.29, 0.75).
A parabola was used in order to make the curvature on the windshield. It is a little compressed as that increases the rate at which the slope increases. As a result, it makes a perfect or close to perfect windshield curvature. I determined the domain restriction by finding the intersection coordinate between equation 15 and 16.
Equation 15 and 16
Equation 15:
Equation 16:
Equation 15 and 16 are linear functions and were used to full small gaps between the functions.
Equation 17, 18, 19
Equation 17:
Equation 18:
Equation 19:
These are horizontal and vertical lines that were used to create the windows and the door line. Equation 17, , was used to make the horizontal component of the windows. Equation 19, , and equation 18, , were used to make the vertical component of the windows and the door. The domain and range were based on the intersection between equation 20 and 21.
Equation 20 and 21:
Equation 20:
Transformations:
• Horizontal translation 0.15 units to the right
• Vertical translation 2.25 units up
• Vertically compressed by a factor of 0.33
• Reflected across the x-axis
This quadratic function has the vertex (0.15, 2.25)
Equation 21
Transformations:
• Horizontal translation 0.6 units to the right
• Vertical translation 2.25 units up
• Vertically compressed by a factor of 0.17
• Reflected across the x-axis
This quadratic function has the vertex (0.6, 2.25)
I wondered if a quadratic function would be useful or a circle when making the windows. A quadratic function was a good choice because of how I wanted the windows to be shaped. A circle would not have been a good choice as it would bend too much. The a value was negative because I wanted the function to open downwards.
Both of the windows are not the same because I wanted to create variation in the car. I wanted the second window (equation 21) to be bigger than the first window, and that’s why I chose to make the a value smaller when compared to equation 20.
Both y values in the vertex of the equations are the same because I wanted to have both of the windows start at the same height. It would look strange to have the windows start at different height. I applied different horizontal translations in both equations so that they fit with the other functions.
Domain restrictions were set in both equations. The restrictions were based x points on the intersection points between the equations 17, 18, and 19 and equations 20 and 21.
Reflection:
Throughout this exploration, I have learned how the different shapes of the functions and relations can be used in order to make an artistic piece and how art doesn’t have to be with pencil, paper or paint. I have learned that math has many artistic elements to it. I enjoyed doing this a lot as I was able to combine 2 of my interest into one topic. I will continue to graph objects as a hobby. In this exploration, I was able to use functions in a way that I have never used before. As I continue on with my passion, I hope to explore even more functions. The possibilities are endless when you incorporate other types of functions such as logarithmic functions, exponentials, sinusoidal and many more. The use of this also has many other benefits. The use of mathematical equations to draw an image can be used in IT. For example, someone may wish to draw a logo of their company using mathematical equations. This is known as vector graphics and has many advantages over raster images. Raster images consist of pixels and when you zoom into a raster image, the quality is lowered, and you are able to see the individual pixels. Vector images use mathematical functions and relations to create the image, so if you zoom in, it looks the same (Vector Conversions, 2018). Moreover, an engineering software known as AutoCAD utilizes mathematical formulas in order to design homes buildings and many more. The use of mathematics to create shape isn’t only used to create pictures but has many uses that affect our daily lives
There were limitations in this exploration. One limitation is that the picture I drew using mathematical equations was not exactly the same as the template I was using. This is a result of the inconsistent change of the curves of the function making it nearly impossible to get the exact curve. My graph was similar to the template but was not exactly the same. Moreover, there were a few small gaps in the car as I was not able to connect some of the functions.
Conclusion:
In conclusion, I was able to successfully meet the goal of this exploration which was to create a car using functions and relations. However, there were some limitations as discussed earlier.
References
Cute Toy Car Coloring Page. Retrieved from http://sweetclipart.com/cute-toy-car-coloring-page-2191
Daimler. Benz Patent Motor Car: The first automobile (1885–1886). Retrieved from https://ww w.daimler.com/company/tradition/company-history/1885-1886.html
Raster vs Vector. Retrieved from https://vector-conversions.com/vectorizing/raster_vs_vector.html
Study.com. Retrieved from https://study.com/academy/lesson/semi-major-axis-of-an-ellipse.html