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Mth 163
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Why do y = x^-2 and y = x^-3 rise more and more steeply as x approaches 0, and why do their graphs approach the x axis as we move away from the y axis?
My Answer:
The x^-2 graph is a mirror image. Each side will have decimal y values because of the negative exponent, and will get steeper as x approaches 0 because the y values will be closer to 1.
The x^-3 graph is split with part of the graph in the first quadrant while the other half is in the third, the y axis providing an asymptote as it did with the x^-2 graph. Again, because the function is a negative exponent, the values will gradually become smaller and smaller fractions as they approach the y axis and are larger and larger as they approach the x axis and 1.
The given solution:
And as x approaches 0 the expressions x^-2 and x^-3, which mean 1 / x^2 and 1 / x^3, have smaller and smaller denominators. As the denominators approach zero their reciprocals grow beyond all bound.
y = x^-2 and y = x^-3 rise more and more steeply as x approaches zero because they have negative exponents they become fractions of positive expressions x^2 and x^3 respectively which have less and less slope as they approach zero. As x^2 and x^3 approach zero and become fractional, x^-2 and x^-3 begin to increase more and more rapidly because their functions are then a whole number; (1) being divided by a fraction in which the denominator is increasing at an increasing rate.
As y = x^-2 and y = x^-3 move away from the y-axis they approach the x-axis because they have negative exponents. This makes them equivalent to a fraction of 1 / x^2 or 1 / x^3. As x^2 and x^3 increase in absolute value, the values of y = x^-2 and y = x^-3 constantly close in on the x-axis by becoming a portion of the remaining distance closer, they will never reach x = zero though as this would be division by zero (since it is a fraction)
Instructor Response to my answer:
y values can get closer to 1 without the graph getting steeper.
When x is 1 the y value is already 1. The graph gets steeper because the y values eventually get very large.
Why is it that the y values get very large?
You also have yet to address why the graphs approach the x axis.
Questions/Answers to response:
After reviewing the given solution, I understand that the reason the graph gets steeper is because the slope becomes less and less as it approaches zero. I also understand that the graph approaches the x axis because the functions have negative exponents, however I don’t understand what is meant by the y values get very large because when you put a number into the x function you get a fraction and fractions are not large numbers. So I’m confused as to what you mean by large numbers.
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@& See if the following additional questions help you make sense of my original questions:
Suppose x = 1/500. What then are the values of x^-2 and x^-3?
Now what if x = 1 / 4 000 000 ? What then would be the values of x^-2 and x^-3?
What does this tell you about the graph of each of these functions?
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Now what do you get if x = 500? And if x = 4 000 000?
What do these results tell you about the graphs?
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