Chapter 3, Polynomial and Rational Functions - Section 3.2 - Polynomial Functions and Their Graphs - 3.2 Exercises - Page 301: 2

a. $y\rightarrow\infty$ as $x\rightarrow\infty$ and $y\rightarrow-\infty$ as $x\rightarrow-\infty$. b. $y\rightarrow-\infty$ as $x\rightarrow\infty$ and $y\rightarrow-\infty$ as $x\rightarrow-\infty$.

End-Behaviour of a Polynomial - Polynomial function of odd degree and positive leading coefficient falls to the left end and rises to the right end. - Polynomial function of odd degree and negative leading coefficient rises to the left end and falls to the right end. - Polynomial function of even degree and positive leading coefficient rises to the left end and right end. - Polynomial function of even degree and negative leading coefficient falls to the left end and right end. a. In this case, $y=x^3-8x^2+2x-15$, the function is of odd degree and positive leading coefficient. therefore, the polynomial function's graph falls to the left end and rises to the right end. thus, $y\rightarrow\infty$ as $x\rightarrow\infty$ and $y\rightarrow-\infty$ as $x\rightarrow-\infty$. b. In this case, $y=-2x^4+12x+100$, the function is of even degree and negative leading coefficient. therefore, the polynomial function's graph falls to the left end and falls to the right end. thus, $y\rightarrow-\infty$ as $x\rightarrow\infty$ and $y\rightarrow-\infty$ as $x\rightarrow-\infty$.