As part of our Grade 9 Extended course we have to cary out a various amounts of investigations. Investigations are there to challenge our brains, because the minimal amount of information is provided.Our current unit is rational functions. In class we have had practiced questions on the unit before we carried out the investigation.
We started off with learning what a rational function is. A rational function is y= (a/(x-b)) + c. By looking at a rational function and simplifying the rational function you are able to find out a lot of information. For example the c number is the horizontal asymptotes. An asymptotes is a line or a curve on a graph, which it approaches but never touches. It is always shown on the graph as a dotted line. In order to find out the vertical asymptotes, one must look at b. B is always the vertical asymptote. The y and the x are the values on the graph.
In order to receive more information, which are the point of discontinuity also known as the holes, the domain and the range. The point of discontinuity is a point on the graph, which is undefined, therefore it is also known as a hole. When you simplify the rational expression you can receive the hole from the expression which you cancelled out. However, you have to make x = 0. The range will never equal the horizontal asymptote. This is shown by writing y/y≠. This means it cannot equal. The domain are always not able to equal the two expressions when you simplified the rational expression. Meaning you would write x/x ≠.
Light it up is a new attraction at a fair. The attraction consists of a laser pointer along a string which is 1.5 meters tall. The laser is on a 10 centimeter platform positioned 25 centimeters from the wall. The point of the game is to try and slide the laser pointer so the bean is reflected off the mirror and hit a target which is 1 meter off the ground. Once you are able to do that then you move onto 3 meters and then 10 meters (see diagram 1 of the game below).
There are tow players which have attempted the game multiple times and are not able to achieve it. They are positive that there must be a mathematical function which shows us the exact position for the laser pointer.
Our mission was to find out the mathematical function also known as the rational function so they are able to win the game. We had to complete these mission with three classmates.
In order to preform this investigation correctly we had to put one tape measure from the wall going up the wall. Then place another on the floor going across the floor. We then placed the mirror 36 centimeters away from the wall. This was our b value as well as our vertical asymptotes. The mirror was 6 centimeters from the floor (on a box, see figure 1), which is the c value and the horizontal asymptotes.The distance the laser pointer had from the floor throughout our investigation which must stay the same was 83 centimeters (see figure 2 and 3 which was our set up as, also look at diagram 2 for a diagram of the set up). It is important all of these stay the same throughout the investigation.
To do the investigation we had to stand on the opposite side of the wall from where the mirror was and aim the laser in the middle of the mirror. Then the image of the laser would be reflected on the wall and then tape the points. We had to decide the x values. In order to get a good graph one must have a wide range. Our ‘x’ values were: 250, 200, 175, 150, 100, 80, 60, 55 centimeters. After we did the investigation from each ‘x’ value all the points we measured each point (see diagram 2, to know how you find out the ‘x’ and the ‘y’ values). We put all of our data in the data table (see table 1)
We then graphed our data, however with our set up as previously mentioned we were already ale to find out the vertical and the horizontal asymptotes. I created the graph and it is proven to be a rational function as well as the asymptotes are correct (see graph 1 below). What the horizontal asymptotes tells us is that the laser pointer is not able to make a reflection on the wall if it is below 6. It is not possible to get data at 6 centimeters due to the fact the mirror was on a 6 centimeters platform ( see figure 5 for the mirror setup). The vertical asymptotes tells us that as soon as you point the laser at 36 centimeters it will go straight up. After 36 centimeters it will go in negative numbers.
In order to get our function we had to solve for a since we did not know this value. We therefore chose a ‘y’ and a ‘x’ value from our data and subsitute it in the equation y= (a/(x-b)) + c. 20 = (a/(250-36)) + 6. In order to solve it one must subtract 6 from both sides and then subtract 36 from 250 which equals 14 = a/214. The next step is to multiply 2014 by 14 which equals 2996. This means a is 2996. Our function is f(x) + (2996/(x-36)) + 6.
This function will help the two people at the fair to be successful at the game, because you can subsitute y, a, b and c in. The reason for this is, because they want to find out what x must be in order. The players want to find out the distance they must stand from the wall if the target is at a certain point. For 1 meter you would subsitute 100 in because 1 meter is 100 centimeters and the whole function is in centimeters. You then solve the equation using algebra. You must do the following for 3 meters from the floor and 10 meters from the floor (see the working out below in figure 5). However if you look book at table 1 you can see that in our results if the laser would be 1 meter away from the wall the laser must be around 80 centimeters from the wall. This shows our results are not accurate. This could be possible, because our function used high b and c values. If we would have used different values it might have been more accurate. One must take into consideration that no investigation is done perfectly. There are always factors, which could affect the result. Some of the factors which could have affected our data, which then made our working out not fit our data was that when one go further away from the wall. It was harder to get the exact middle of the mirror. One also shakes the laser, therefore, it is not always kept straight making it hard to get the exact measurements. Another factor could be that the chair was not exactly 250 centimeters from the wall due to the fact the chair goes up side ways. In order to point the laser you also put it on top of the chair on a bit forwards. This makes the ‘x’ value inaccurate. A factor which could lead to the people playing the game not hitting the target is if one does not point the laser to the exact middle of the mirror. In order to hit the target one must point the laser exactly on the middle. Another factor is they way one holds a laser. As well as they was one stands from the distance. If one puts their arm extremely out and does not use the exact measurements they will not hit the target.
In order for the players to be successful at the game one must be precise, have patience, and not be shaky. It is a very mathematical game but if you have the right measurements one must be able to hit the target.