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This particular mathematics graph interpretation will discuss how graph of differs from and. The purpose of this project is to study the relationship among those functions.

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Introduction

Introduction

        This particular mathematics graph interpretation will discuss how graph of image26.png differs from image27.png andimage32.png. The purpose of this project is to study the relationship among those functions. The guiding question is: How do they differ from one another and what patterns do they have?

Hypothesis

        The functionimage27.png will differ from the other functionimage26.png by having all the parts of the functionimage26.png below the x-axis, where negative units lie for y – axis, as a horizontal reflection. Because absolute value can only have positive value as a result, the function will also have positive value only. Therefore, when having an absolutely value graph, because only positive value exists, the negative value part of the graph will reflect upon x –axis. In addition, the original graph’s y – intercept and range will have positive sign of the original value and be equal to or greater than zero respectively for absolute value function.

image00.png

image12.pngimage01.png

Comparing image26.png and image27.png

Absolute value graph of image26.png and image27.png

image20.pngimage19.png

image22.pngimage21.png

image24.pngimage23.png

image25.pngimage02.png

General Statement

When changing the equation from image26.png toimage27.png, the bottom part (the part where y has a negative value)

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Middle

 and image30.pngimage03.png

image04.png

image05.png

image31.png

image33.pngThe graph above is when the graph of image34.png and image35.pngwhich is drawn in bold line are drawn together. By looking at the graph, it is clear that the function ofimage36.png’s negative y-value part reflects upon x-axis to form the graph of absolute value. It makes x-intercepts, domain and asymptote for both of the function to remain the same while the y-intercept changes its sign because of the absolute value. However, if the graph had asymptotes, the function of absolute value and the original function will have the same asymptote except that the asymptote for function of absolute value will only work in 2nd and 3rd quadrants. Also, because it has done horizontal reflection, the range becomesimage37.png. Therefore, the conjecture is always true and any function will obey this pattern.

Drawingimage27.png graph on given graph of image26.png

image38.png

Comparing image26.png and image32.png

Hypothesis

Unlike the relationship between image26.png and image27.png graphs, the functionimage32.png will differ from the other functionimage26.png by having all the left part of the functionimage26.png about y-axis, where negative units lie for x – axis, as vertical reflection.

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Conclusion

image43.png: the part of the graph of 1st and 4th quadrant of image42.png reflects upon y-axis. It makes x-intercept, domain and asymptote to change and y – intercept to remain the same. Furthermore, because the function of absolute value has done vertical reflection, the range will start from the minimum or maximum point of the 1st and 4th quadrants of the graph. Therefore, the conjecture is always true and any function will obey this pattern.

Drawing image32.png graph on given graph of image26.png

image45.png

Conclusion

        After comparing these functions, it was noticeable that an absolute value makes significant differences in the graphs. When comparing image26.png and image27.png, it was found that the negative y-value part of the graph reflects upon the x-axis because an absolute value creates only positive values, and negative values can never exist. When comparing image26.png andimage32.png, it was found that the positive x-value part of the graph reflects upon the y-axis, because the absolute value of “x” always creates the positive value of “y” regardless of x-value’s sign. Therefore, these functions differ from each other, and the function of the absolute value of either the whole function or the x-value only, each has its own specific patterns.

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