11.3.2. End behavior

The end behavior describes how a graph appears as the independent variable approaches infinity to the right (x increases) or to the left (x decreases). It depends whether the degree of the polynomial is odd or even and the sign of the coefficient of the highest order term, a_n. The end behavior for all possible cases is:

Sign of an	Degree	End behavior
positive	even	Approaches positive infinity as x decreases and as x increases.
positive	odd	Approaches negative infinity as x decreases and approaches positive infinity as x increases.
negative	even	Approaches negative infinity as x decreases and as x increases.
negative	odd	Approaches positive infinity as x decreases and approaches negative infinity as x increases.

Figure 11-3: Cubic function

f(x) = x3 - 3x2 - x + 3

{-1.000, 0.000}, {+1.000, 0.000}, {+3.000, 0.000}

{-0.155, +3.079}   local maximum
{+2.155, -3.079}   local minimum

Figure 11-4: Quartic function

f(x) = x4 - 5x2 + 4

{-3.000, 0.000}, {-1.000, 0.000}, 
{+1.000, 0.000}, {+2.000, 0.000}

{-1.581, -2.250}   local minimum
{ 0.000, +4.000}   local maximum
{+1.581, -2.250}   local minimum

Figure 11-5: Quintic function

f(x) = x5 - 5x3 + 4x

{-3.000, 0.000}, {-1.000, 0.000}, {0.000, 0.000}, 
{+1.000, 0.000}, {+2.000, 0.000}

{-1.644, +3.631}   local maximum
{-0.544, -1.419}   local minimum
{+0.544, +1.419}   local maximum
{+1.644, -3.631}   local minimum