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Function Transformation Investigation

Extracts from this document...

Introduction

Math: Graph Transformations

Sebastian

image00.pngimage01.png

image04.jpg


Graph Transformations

Note: The arguments of trigonometric functions are done in radians.

        A function, in mathematics, is an expression that, from an input number, outputs another number that is nearly always influenced by the input one. An interesting attribute to a function is that it can be graphed on a plane. A graph is represented on a 2d plane or Cartesian plane when the function has 1 input variable. Functions can also have more than 1 input variable, but are not graphed on a 2d plane. For example a graph with two input variables is represented on a 3d plane. Only functions with 1 input variable (therefore represented on a 2d plane) will be examined.

To understand how a function is graphed, let’s have a look at these common functions and their corresponding graphs:

image05.pngimage78.png

Graph of image102.pngimage102.png                        Graph of  image75.pngimage75.png

image121.pngimage130.png

Graph of image06.pngimage06.png                         Graph of image13.pngimage13.png

image26.pngimage33.png

Graph of image40.pngimage40.png                         Graph of image53.pngimage53.png

        Here are some graphs of more complicated functions:

image70.pngimage79.png

Graph of image85.pngimage85.png                          Graph of image99.pngimage99.png

        The last two graphs are similar in shape to previous graphs. The first graph has the same shape as the previously seenimage62.pngimage62.png. The only noticeable difference is that a translation of image100.pngimage100.png has been applied to the first graph producing the graph ofimage101.pngimage101.png.  We can see this by comparing the two graphs on the same plane:

image103.pngimage104.png

The blue graph is image06.pngimage06.png and the red one isimage101.pngimage101.png.

(-5; 0) – (0; 0) = (-5; 0) and (-8; 9) – (-3; 9) = (-5; 0)

We can verify this at larger scales, to ensure that this is effectively a translation and not a translation with a minute stretch, which only starts to have a visible impact at larger scales.

(12; 289) – (17; 289) = (-5; 0)

5 is significant because it is what was added to x before the result was squared up. So we could conclude that for any function image105.pngimage105.png its translation compared to image06.pngimage06.png is (-a; 0).

...read more.

Middle

        These two graphs not only confirm that multiplying the output or input stretches the graph but it also helps us understand how to calculate the stretch and also the direction.

        The first graph (cosine graph) that multiplies the input shows us that a horizontal stretch occurs. The stretch is seen as beingimage132.pngimage132.png, because the input was multiplied by 3. This observation is made by looking at the intervals difference (circled in black) between the original graph and the transformed one. The stretch happens along the y axis.

        The second graph represents a sine graph and its transformation done by multiplying the output.  The stretch is a vertical one. The two circled points help us notice that it is a stretch of ½, because the output was multiplied by 2. It is along the x axis that the stretch occurs.

Why did this happen? For example, let’s say that the input of a function was multiplied by 3. Arriving to an output of 5 will take  image133.pngimage133.png of the original distance, because the input value was multiplied by 3.  This creates a compression (or stretch) horizontally, because that is where the input values are plotted. The explanation of what happened to the graphs where the output was multiplied is simply that whatever was outputted was multiplied by the inverse of the multiplication factor, therefore stretching (or compressing) the graph vertically. Multiplying (or dividing) by a negative number reflects the graph. (More about reversing the sign of a number [it is equivalent to multiplying it by -1] is available later in the investigation.)

        Another way to explain this transformation is the following:

image134.png

image135.png

image136.png

        This shows how the output ends up being divided (or multiplied by the inverse).

  • We now know that a graph is stretched when its output or input is multiplied.

...read more.

Conclusion

        Sure enough, the results are as expected:

image84.png

        Another transformation that did not make a function because it resulted as a one-to-many equation was still noteworthy of exploration. What was changed to the input or output was adding a ± sign before them.

image86.pngimage87.png

image88.pngimage88.pngimage89.pngimage89.png

image90.pngimage91.png

image92.pngimage92.pngimage93.pngimage93.png

These graphs just create a mutual reflection in x and y depending whether the input or the output was changed.

        Changes such as these are easily explainable; in the cases ofimage94.pngimage94.png, positive and negative values of x(input) are considered, just on the right side of the quadrant (where x is positive). The same is true for the left side (where x is negative), therefore creating this mutual reflection in the y axis. Putting ± as a multiplier of the output plots the graph in its positive and negative values, creating this mutual reflection along the x axis. As a quick note, we can also notice that the rule of the output modification creating a vertical transformation and the input modification creating a horizontal transformation also applies in these graphs.

Up till now, only graphs of function have been explored. Equations also can de graphed and modified. For example, we can graph (x−a)²+(y−b)²=r².

In the following graph, a=0; b=0; r=1

image95.png

a=2; b=1; r=1

image96.png

a=2; b=1; r=3

image97.png

From these observations, a and b change the x and y coordinate of the circle and r changes its radius.

If we think about it, it is actually logical.

image98.png

        Since each variable is squared, this means that it is a one-to-many equation. Therefore multiple points will be plotted because there are two solutions for each values. This is what forms the circle. a and b are what creates the circle shift; which is somewhat similar to earlier observations. Finally r changes the diameter, because when changed, it forces x and y to adapt to the change, creating a bigger or little circle, depending on the range of the solutions.

image03.png

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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