I have to determine the end behavior of the graph of each polynomial function. I'm just really confused on how to graph variables with exponents..

Question

anonymous · Answer

$$y=x ^{2}-2x+3$$

$$y=\frac{ 1 }{ 2 }x^{4}+5x ^{2}-\frac{ 1 }{ 2 }$$

whpalmer4 · Answer

Same way you would graph variables without exponents.  You plug in the number for $x$  and find the corresponding value for $y$.  

In your first equation, $y = x^2-2x+3$

$x = 0, ~y = (0)^2-2(0)+3 = 0-0+3$
$x=1,~y = (1)^2 - 2(1)+3 = 1-2+3 = 2$
$x=2,~y = (2)^2-2(2)+3) = 4-4+3=3$
etc.  You should also plot some points with negative values of $x$.

whpalmer4 · Answer

But to find end behavior, you don't really need to graph anything — just look at the term with the largest exponent and decide what it does at very positive and very negative values.  For example, if our function happened to be $y = 3x^3 + 2x^2 -5x + 4$, at large values of $x$ (both positive and negative) the $3x^3$ term is going to dominate the result.  Consider $x=100$: the respective terms are $3(100)^3, 2(100)^2,-5(100),4$ or $3000000, 20000,-500,4$  As $x$ gets closer to positive or negative infinity, the value will be almost entirely that of the $3x^3$ term.

So, for my example (which is NOT your problem), the end behavior is down and to the left to negative infinity at negative values of $x$, and up and to the right to positive infinity at positive values of $x$, because that is the end behavior of $3x^3$.  Plug in a big positive number, you get a big positive number in return.  Plug in a big negative number, you get a big negative number in return.

Any questions?