So, I was playing with an online graphing calculator and I plugged in 0.5x^{2 }x^{2 }and 2x^{2 }and when I went up to higher numbers, such as x=10, I noticed they were unevenly spaced. This puzzles me because 0.5x^{2 }and 2x^{2 }SHOULD be equally spaced from x^{2 }because they are both off by a factor of 2, but 0.5x^{2 }is farther away from x^{2 }than 2x^{2 }is. :O
Actually, the only point where 0.5x^{2} and 2x^{2} are evenly spaced from x^{2} is at x = 0.
This is found by taking the difference between x^{2} and the other equation at any given point
|2x^{2} - x^{2}| and |0.5x^{2} - x^{2}|
And then setting them equal to each other.
|2x^{2} - x^{2}| = |0.5x^{2} - x^{2}| → x^{2} = 0.5x^{2} → x = 0.5x
And then setting them equal to each other.
|2x^{2} - x^{2}| = |0.5x^{2} - x^{2}| → x^{2} = 0.5x^{2} → x = 0.5x
But that isn't your question. Numbers always have patterns, so let's see if we can find this one. :)
Using the same functions, we can make a table to look at how things are spaced. I'll be going up in increments of 1, starting at 0.
0.5x^{2} | x^{2} | 2x^{2} | 2x^{2} - x^{2} | x^{2} - 0.5x^{2}
Using the same functions, we can make a table to look at how things are spaced. I'll be going up in increments of 1, starting at 0.
0.5x^{2} | x^{2} | 2x^{2} | 2x^{2} - x^{2} | x^{2} - 0.5x^{2}
0 | 0 | 0 | 0 | 0
0.5 | 1 | 2 | 1 | 0.5
2 | 4 | 8 | 4 | 2
4.5 | 9 | 18 | 9 | 4.5
8 | 16 | 32 | 16 | 8
So there's our pattern. Comparing just the last two columns, we see that for each increment the difference is doubled. This makes sense because the difference between 2x^{2} and x^{2} is twice as large as the difference between 0.5x^{2} and x^{2}.
I hope that helped!
I hope that helped!
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