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

Bruce B. | Mathematics and Physics; Calculus and Mechanics SpecialtyMathematics and Physics; Calculus and Me...

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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

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.

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}.

Dennis R. | Math Tutor, Grades 4 - 12, Math Certified, 47 Yrs experienceMath Tutor, Grades 4 - 12, Math Certifie...

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The coefficient in front of the x squared causes the graph to change the width of the graph.

The coefficient of 1 is the basic x squared function. If the coefficient is above 1, then the graph will rise faster and be narrowed. If the coefficient is less than 1 then the graph will rise slower and thus be a wider graph. foe instance if x is 3, then x^2 = 9, .5 * x^2 = 4.5, and 2 * x^2 = 18.

If you have a graphing calculator, use the Table (2nd Graph) to see all three functions compared for each value of x.

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