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Transforming y=1/x
QUESTION: What would happen if we changed the equation to y = 1/(x +/- k), where k is some constant integer?
For our sake, lets make k = 5. How do you think this would change our original graph of y = 1/x? Let us graph y=1/x, y=1(x+5), and y=1(x-5).
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Do you notice anything about the graphs? Did they change shape? Did they move?
Notice |
Do the graphs look any different? You should notice that the graphs shift units left and right. When the constant k is on the inside and it is added, the basic graph shifts left k units. When k is subtracted from basic graph then the basic graph is shifted right k units. |
| Notice the Asymptotes | The original asypmtotes were at x = 0 and y = 0. Notice that when the basic graph is shifted k units left or k units right, that the vertical asymptote shifts k units in the same direction. The horizontal asymptote stays the same. |
Below is all of the equations graphed together. This may help you see the changes of adding and subtracting a constant to the variable itself.
| The graph to the right shows the graphs of y = 1/x, y = 1/(x+5), and y = (x-5). |  |
| NOTICE: | The blue line represents the original graph of y=1/x. You should be able to see that the green line is shifted to the right of the blue line. The red line is shifted to the left of the blue line. This is because of properties of transformations. The addition and subtration to the x values within the parenthesis shifts the original graph to the left and the right k units with k being the constant. If subtracting, like with the green line, then shifts the graph to the right. If adding, like the red line, then shifts the graph to the left. |
| NOTICE: | Notice how the graphs each try to reach zero but never actually hit zero. This is because the equations can never equal zero. |