The end behavior of a polynomial is determined entirely by the leading term. For example, [tex]2x^4-x+1[/tex] behaves the same way as [tex]2x^4[/tex] when [tex]x[/tex] is very large in magnitude. All the problems in the picture are more or less the same. I'll pick out two examples:
1a) [tex]f(x)=5x^6-3x^3-4x^2+8x[/tex] behaves like [tex]5x^6[/tex]. [tex]x^6[/tex] will be positive regardless of the value of [tex]x[/tex], so to either the left or right, the graph of [tex]f(x)[/tex] will approach positive infinity or "rise" on both the left and right sides.
1c) [tex]f(x)=-4x(x-4)(x+2)[/tex] has a leading term of [tex]-4x^3[/tex], so [tex]f(x)[/tex] behaves like [tex]-4x^3[/tex]. When [tex]x<0[/tex], we have [tex]x^3<0[/tex], so that [tex]-4x^3>0[/tex]. This means [tex]f(x)[/tex] rises to the left. If [tex]x>0[/tex], then [tex]-4x^3<0[/tex], so [tex]f(x)[/tex] falls to the right.
