Changing the equation from to
When we have a graph of y =, we take the absolute value of y. If it is positive, it leaves the number unchanged; absolute value of four is just four. Therefore, wherever f(x) is positive, the graph of appears the same. However, the absolute value of a negative number is the number itself without the sign; absolute value of -4 is 4, which you can think as – (-4). So wherever f(x) is negative, you can regard as –f(x). For a given value of “x”, if “y” is negative, “y” can be replaced with –y, making it positive. Therefore, absolute value over the whole function will have the part below the x-axis in the graph to fold up to be above the x-axis.
Even though some graphs that do not include negative y-values will not look as if they obey this pattern, every function is going to obey these patterns. Therefore, when a function has an absolute value, the graph will be reflected vertically upon x-axis.
Conjecture
and
The graph above is when the graph of and which is drawn in bold line are drawn together. By looking at the graph, it is clear that the function of’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 becomes. Therefore, the conjecture is always true and any function will obey this pattern.
Drawing graph on given graph of
Comparing and
Hypothesis
Unlike the relationship between and graphs, the function will differ from the other function by having all the left part of the function about y-axis, where negative units lie for x – axis, as vertical reflection. Because absolute value of “x” can only have positive value of “x”, as a result, the function will also have positive value of “x” only. Therefore, when having an absolute value of “x” graph, because only positive value of “x” exists, the positive value part of the graph will reflect upon y –axis so that the graph is symmetrical to y – axis. Furthermore, original graph’s x – intercept will have the same value but different sign such as and range will remain the same as the original graph’s range.
Absolute Value Graphs - and
General Statement
When changing the equation from to, the graph on the right side reflects upon y-axis. Also, in the graph, x-intercept, domain and asymptote change except for certain circumstances such as having no x-intercept, and y – intercept remains the same. Furthermore, the absolute value of “x” value function has the range from the minimum value of the positive “x” value side. (1st and 2nd quadrants’)
Changing the equation from to
Because the absolute value is on “x” and it makes every x-value positive, it creates both positive and negative “x” values to have the same y-value. For an example, if. Because is an absolute value, x can be either 4 or -4. Whatever the case is, the y-value is not going to change from 1. Therefore, whichever value “x” has, “y” value is going to remain the same for both positive case and negative case.
Every function is going to obey this pattern however the x-intercepts, y-intercepts, domain, range, and asymptote may vary depending on the graph. When the graph does not have x-intercept or asymptote but has absolute value of x in the function, x-intercept and asymptote of both graphs will not change from “does not exist” or “unidentified”.
Conjecture
and
The graph above is when the graph of and which is drawn in bold line are drawn together. The graph gives clear interpretation of how the function of transform the function of: the part of the graph of 1st and 4th quadrant of 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 graph on given graph of
Conclusion
After comparing these functions, it was noticeable that an absolute value makes significant differences in the graphs. When comparing and , 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 and, 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.
