Whats up with spacing of x^2, 0.5x^2, and 2x^2!?

Question

So, I was playing with an online graphing calculator and I plugged in 0.5x² x² and 2x² and when I went up to higher numbers, such as x=10, I noticed they were unevenly spaced. This puzzles me because 0.5x² and 2x² SHOULD be equally spaced from x² because they are both off by a factor of 2, but 0.5x² is farther away from x² than 2x² is. :O

Answer 1

Actually, the only point where 0.5x² and 2x² are evenly spaced from x² is at x = 0.

This is found by taking the difference between x² and the other equation at any given point

|2x² - x²| and |0.5x² - x²|

And then setting them equal to each other.
|2x² - x²| = |0.5x² - x²| → x² = 0.5x² → 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.

0.5x² | x² | 2x² | 2x² - x² | x² - 0.5x²

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 2x² and x² is twice as large as the difference between 0.5x² and x².

I hope that helped!

 

Answer 2

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.

Hope this is helpful.