# Functions Culminating Task

Grade 11 Functions Culminating Task

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Google Trends (https://trends.google.com/trends/) is an online tool that allows a user to search the global interest of a “search-term” over time. The y-axis shows the relative number of searches and the x-axis shows time (the scale can be changed).

For example:

The image to the right shows the search interest for the film Star Wars: The Last Jedi throughout the week of its release. The first arrow shows low interest by comparison to the rest of the graph. The second arrow demonstrates the highest point of interest, 11 pm on the day of its release (Dec 15th, 2017). The third arrow indicates slowly decreasing interest in the film, but still increasing to a peak at 11 pm each evening following the film’s release.

INSTRUCTIONS – USING GOOGLE TRENDS AND DESMOS

The following instructions can also be found at this video link: https://youtu.be/yyskaUPz-X4

1. Think of an example of a real-world “search-term” that would produce an “interest over time” graph that includes as many functions that you have learned about in this course as possible. Possible functions include but are not limited to exponential growth, exponential decay, sinusoidal functions, etc. (see figure above)

1. Type your chosen search-term in the search bar. Press enter and view the graph that is generated. You may need to adjust the scale on the x-axis to obtain a more interesting graph depending on your chosen search-term.

1. Once you are satisfied with your graph, take a screen shot.

1. Insert your screenshot into Desmos (https://www.desmos.com/calculator) using the insert menu (shown below in the image on the right).

1. Reposition your image in such a way that makes sense (ie. if your function has a horizontal asymptote, have it sit on y=0 as show below). For the graph shown below, the scale is set so that the maximum occurs at 5 units on the Desmos grid.

Your Desmos workbook is now set up and you are ready to begin your culminating task!

PART A – MODELING YOUR TREND WITH FUNCTIONS

An example can be found at this link: https://youtu.be/HUrBGV0PD0c

Using your Desmos workbook, begin plotting base functions and transforming them to attempt to model your trend. Do this by starting with an appropriate base function (ie. y = 2x) and applying your understanding of transformations.  See the link above for assistance with this.

Apply your transformations one at a time, taking screenshots and pasting them into the chart below. Fill in the equation of your transformed function and your rationale for each transformation. Your goal is to show your progress as you develop your model. You may add rows to the chart as needed.

 Screenshots Equation Rationale [Paste your screenshot of your original trend here]

PART B – ALGEBRAIC CALCULATIONS

You now have a model that relates the interest level of your search-term (y-axis) to the date (x-axis).

1. Determine how many units on Desmos represent a unit of time (x-axis) and a unit of interest (y-axis). This will be necessary for interpreting your model in a real-world context.
2. Choose a random date on the x-axis. Using your final transformed equation that you developed, determine the interest level of your search-term at that given day.
3. Compare your result to what the interest level was on both your model and the original trend you obtained from Google Trends. Which is your result closer to? What does this say about the accuracy of the model you developed? What does this say about mathematical models in general?

PART C – ANALYSIS OF YOUR TREND

1. What types of functions did you include to model your Google Trend image? Why did you choose the functions that you did?
2. Conduct research (and think about your own personal memories if applicable) into what was going on in the world with respect to your search-term at the significant points on your graph (ie. maximum points, points of major fluctuation, points where the shape changes significantly). Is there a relationship between what was happening in the world and what you are seeing on your graph? Discuss your findings and explain, with reference to your search term, why you think the trend behaves the way it does during each time period.

For each of the transformed functions that you used, describe the domain, range and interpret and explain them using th

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