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Visually Representing Numerical Data | Splitting up the Whole with Pie Graphs | Showing the overlap with Venn Diagrams | Comparing Categories with Bar Graphs | Seeking Trends with Line Graphs | Showing Orderly Data with Histograms | Functions as Graphs in the Coordinate System | Summary Functions as Graphs in the Coordinate System
In Lesson 13, you learned that a function is a value whose size depends on the size of another value. If you think about Line Graphs and Histograms, they are functions. The one value is the size of the bar or the height of the line, and it depends on the other value, which is the value of the item in the interval. Specifically, if you pick any value on the horizontal line, any value that falls within the intervals you are working with, you can then measure upwards until you hit the top of the bar or run into the line. So the input to the function is the category value and the output from the function is the height. The idea of a function being the rule that gives a particular output when something is input can lead us to a picture. Set up a horizontal axis and a vertical axis and use the horizontal one as inputs for the function and use the vertical one for the output of each horizontal input. Usually, the unknown for the input is called X and the unknown for the output is called Y. The mathematical terminology for the following diagram is "Coordinate Axes" where the value of the horizontal position is called the "X coordinate" and the value of the vertical position is called the "Y coordinate" of the points being graphed.
An example of a function would be to create the output by taking half of the input value and adding 1. First, lets compute a table of values by considering some of the possible inputs and computing the resulting outputs. Note that the symbols for "take half the input" is X/2 so the algebra that describes the function is X/2 + 1. More formally, the output is called Y, so the so-called equation of the function is Y = X/2 + 1. That is what we will use to get a table of values then eventually graph.
The graph is done by putting a dot for each point of data, X on the horizontal and Y on the vertical, and connecting the dots left to right.
All other graphs of functions follow the same idea – they are fundamentally just a table of values that have been drawn on the coordinate axes and then connected with a smooth curve. For example, the sine and cosine and tangent data from Lesson 13 can be used to create their graphs. The graph of the sine curve is as follows:
Another type of graph commonly seen in Algebra is a parabola – that’s the shape made by something with X squared, X, and constants in its equation. For example, the graph of the function Y = X2 + X + 1 looks like
Now let's do Practice Exercise 17-4 The four visual representations most commonly used by the media each have their particular strengths and uses. Pie Graphs split up a whole. Bar charts illustrate differences between categories. Line Graphs suggest time-related activity. Histograms tally up frequencies. Each can also be manipulated to show the data in ways designed to influence the viewer, usually by fiddling with the scales on which the data is displayed. Two mathematical visual representations were also shown. Venn Diagrams show us how categories can overlap, while graphs of functions provide a visual way of representing the algebra of a function. Please go to the link below for additional instruction for this topic included on the HSEE: Modeling Real World Situations with Functions:
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