Question

Analyze the end behavior of each function below. Then, choose one of the functions, and explain how you determined
the end behavior.
a. f(x) = x4
b. g(x) = −x4
c. h(x) = x3
d. k(x) = −x3

Answer 1

Answer:

End behavior of a polynomial function depended on the degree and its leading coefficient.

1. If degree is even and leading coefficient is positive then

$p(x)\rightarrow \infty\text{ as }x\rightarrow \infty$

$p(x)\rightarrow \infty\text{ as }x\rightarrow -\infty$

2. If degree is even and leading coefficient is negative then

$p(x)\rightarrow -\infty\text{ as }x\rightarrow \infty$

$p(x)\rightarrow -\infty\text{ as }x\rightarrow -\infty$

3. If degree is odd and leading coefficient is positive then

$p(x)\rightarrow \infty\text{ as }x\rightarrow \infty$

$p(x)\rightarrow -\infty\text{ as }x\rightarrow -\infty$

4. If degree is odd and leading coefficient is negative then

$p(x)\rightarrow -\infty\text{ as }x\rightarrow \infty$

$p(x)\rightarrow \infty\text{ as }x\rightarrow -\infty$

(a)

$f(x)=x^4$

Here, degree is even and leading coefficient is positive.

$f(x)\rightarrow \infty\text{ as }x\rightarrow \infty$

$f(x)\rightarrow \infty\text{ as }x\rightarrow -\infty$

(b)

$g(x)=-x^4$

Here, degree is even and leading coefficient is negative.

$g(x)\rightarrow -\infty\text{ as }x\rightarrow \infty$

$g(x)\rightarrow -\infty\text{ as }x\rightarrow -\infty$

(c)

$h(x)=x^3$

Here, degree is odd and leading coefficient is positive.

$h(x)\rightarrow \infty\text{ as }x\rightarrow \infty$

$h(x)\rightarrow -\infty\text{ as }x\rightarrow -\infty$

(d)

$k(x)=-x^3$

Here, degree is odd and leading coefficient is negative.

$k(x)\rightarrow -\infty\text{ as }x\rightarrow \infty$

$k(x)\rightarrow \infty\text{ as }x\rightarrow -\infty$