EXPLORATION 1 Investigating the End Behavior of Graph each function in the window by

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

EXPLORATION 1 Investigating the End Behavior of

\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{a}_{\boldsymbol{n}} \boldsymbol{x}^{\boldsymbol{n}}

Graph each function in the window

[-5,5]

[-15,15]

. Describe the end behavior using

\lim _{x \rightarrow \infty} f(x)

and

\lim _{x \rightarrow-\infty} f(x)

. 1. (a)

f(x)=2 x^{3}

(b)

f(x)=-x^{3}

(c)

f(x)=x^{5}

(d)

f(x)=-0.5 x^{7}

(a)

f(x)=-3 x^{4}

(b)

f(x)=0.6 x^{4}

(c)

f(x)=2 x^{6}

(d)

f(x)=-0.5 x^{2}

(a)

f(x)=-0.3 x^{5}

(b)

f(x)=-2 x^{2}

(c)

f(x)=3 x^{4}

(d)

f(x)=2.5 x^{3}

Describe the patterns you observe. In particular, how do the values of the coefficient

a_{n}

and the degree

n

affect the end behavior of

f(x)=a_{n} x^{n}

? (1) (a)

f(x)=2 x^{3}

(2) (b)

f(x)=-x^{3}

(3) (c)

f(x)=x^{5}

(4) (d)

f(x)=-0.5 x^{7}

(5) (a)

f(x)=-3 x^{4}

(6) (b)

f(x)=0.6 x^{4}

(7) (c)

f(x)=2 x^{6}

(8) (d)

f(x)=-0.5 x^{2}

(9) (a)

f(x)=-0.3 x^{5}

(10) (b)

f(x)=-2 x^{2}

(11) (c)

f(x)=3 x^{4}

(12) (d)

f(x)=2.5 x^{3}

(13) Describe the patterns you observe. In particular, how do the values of the coefficient

a_{n}

and the degree

n

affect the end behavior of

f(x)=a_{n} x^{n}