Understanding the Transformation of a Graph - Can You Help Clarify?

[bjgawp]

http://img147.imageshack.us/img147/6277/1fox7.jpg

For this graph, I'm not sure exactly how the equation y = (x³+1)² is transformed to get the graph (x³/8 + 1)². I said that it was just horizontally stretched by a factor of 8 but there has been a change in its intercepts. Also, not all the x-coordinates are changed by a factor of 8. Can anyone help clarify this for me? Thanks in advance

 

[0rthodontist]

Compare $$(x^3 + 1)^2$$
to $$(\frac{x^3}{8} + 1)^2$$
The latter equals
$$((\frac{x}{2})^3 + 1)^2$$
Now let's say that I plug in a value x₁ to f(x) for some function f, and get a value y₁ back. Now what number x₂ should I plug into the function g(x) = f(x/2) to get the same value y₁? This is asking, what should x₂ be if you have f(x₁) = f(x₂ / 2)? Do you see how to apply this to your problem?

 

[bjgawp]

Oh .. so it isn't horizontally stretched by a factor of 8 but rather a factor of 2? It seems to work out that way:

f(x) - (1,4), (-1,0) ...
g(x) - (2,4), (-2,0) ...

Hmm... can someone explain why is it so? At a first glance, one would assume that the 1/8 would affect the original graph rather than 1/2

 

[0rthodontist]

Hmm... can someone explain why is it so?

 

[HallsofIvy]

As was just pointed out to you, g(x) = f(x/2). x itself is not divided by
8. That should be all the expanation you need.