Question

use symmetry to sketch the graph of y=x^3+3

Answer 1

\[y=x^3+3\]
Let's find the symmetry:
The peculiarity of this function is that the only independent variable ( x ) is raised to an odd exponent. Note what happens we have only x^3, if you plug in a POSITIVE value 'a' you get
\[a^3\]
but if you plug in '-a' you get
\[-a^3\]



We could describe this behavior mathematically like this:
\[f(x)=-f(-x)\]

Back to our function:
All this means is that, once you've drawn it for the positive values of x, for the negative values you just have to hang yourself to the ceiling and draw the same piece of function again! Another way is to draw the same piece of function, but upside down, because now you are taking the opposite values of y.

But wait... what about that +3 ? '+3' adds to ALL the values computed by x^3, so this means that the whole graph is moved up by 3 units. (by a vector (0;+3)
If there had been no +3 the function would have been symmetrical with respect to the origin of the axes, since it is all moved up by 3, your function will be symmetrical with respect to point (+3;0)

All clear?

Answer 2

One more hint:
this is approximately how the graph of x^3 only looks like!

Note how for the point that have negative values of x, the absolute value for y is the same (same distance to the x axis), but with opposite sign!