Question

1. describe the end - behavior for each polynomial function (use limit notation).

a. $g(x)=4x^{7}-3x^{4}+x$ as $x$ decreases without bound: as $x$ increases without bound:
b. $p(x)=7x^{4}+3x^{3}-3x - 4$ as $x$ decreases without bound: as $x$ increases without bound:
c. $f(x)=-7x^{9}-8x^{3}+6$ as $x$ decreases without bound: as $x$ increases without bound:
d. $g(x)=-3x^{6}+5x^{3}-2x + 6$ as $x$ decreases without bound: as $x$ increases without bound:
e. $f(x)=x^{3}+4x^{2}-3$ as $x$ decreases without bound: as $x$ increases without bound:
f. $p(x)=-8x^{2}-3x + 10$ as $x$ decreases without bound: as $x$ increases without bound:

1. the graphs, equations, and limit statements for four polynomial functions are below. match each graph and equation to the corresponding limit statement.

i. $lim_{x
ightarrow-infty}f(x)=infty$, $lim_{x
ightarrowinfty}f(x)=infty$ equation: graph:
ii. $lim_{x
ightarrow-infty}g(x)=infty$, $lim_{x
ightarrowinfty}g(x)=-infty$ equation: graph:
iii. $lim_{x
ightarrow-infty}h(x)=-infty$, $lim_{x
ightarrowinfty}h(x)=infty$ equation: graph:
iv. $lim_{x
ightarrow-infty}k(x)=-infty$, $lim_{x
ightarrowinfty}k(x)=-infty$ equation: graph:
a. $x^{3}+bx^{2}+cx + d$
b. $-\frac{1}{20}x^{4}+bx^{2}+d$
c. $\frac{1}{20}x^{4}+bx^{2}+d$
d. $-\frac{1}{4}x^{3}+bx^{2}+d$

Explanation:

Step1: Recall end - behavior rules for polynomials

The end - behavior of a polynomial \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_0\) is determined by the leading term \(a_nx^n\), where \(n\) is the degree of the polynomial and \(a_n\) is the leading coefficient. If \(n\) is even and \(a_n>0\), \(\lim_{x
ightarrow\pm\infty}f(x)=\infty\). If \(n\) is even and \(a_n < 0\), \(\lim_{x
ightarrow\pm\infty}f(x)=-\infty\). If \(n\) is odd and \(a_n>0\), \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=-\infty\). If \(n\) is odd and \(a_n < 0\), \(\lim_{x
ightarrow\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=\infty\).

Step2: Analyze function A: \(g(x)=4x^7 - 3x^4+x\)

The leading term is \(4x^7\) (degree \(n = 7\), odd; leading coefficient \(a_n=4>0\)). As \(x
ightarrow-\infty\), \(\lim_{x
ightarrow-\infty}g(x)=-\infty\) since for an odd - degree polynomial with positive leading coefficient, the function goes to negative infinity as \(x\) decreases without bound. As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}g(x)=\infty\).

Step3: Analyze function B: \(p(x)=7x^4 + 3x^3-3x - 4\)

The leading term is \(7x^4\) (degree \(n = 4\), even; leading coefficient \(a_n = 7>0\)). So \(\lim_{x
ightarrow-\infty}p(x)=\infty\) and \(\lim_{x
ightarrow\infty}p(x)=\infty\).

Step4: Analyze function C: \(f(x)=-7x^9 - 8x^3+6\)

The leading term is \(-7x^9\) (degree \(n = 9\), odd; leading coefficient \(a_n=-7<0\)). As \(x
ightarrow-\infty\), \(\lim_{x
ightarrow-\infty}f(x)=\infty\) and as \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}f(x)=-\infty\).

Step5: Analyze function D: \(g(x)=-3x^6 + 5x^3-2x + 6\)

The leading term is \(-3x^6\) (degree \(n = 6\), even; leading coefficient \(a_n=-3<0\)). So \(\lim_{x
ightarrow-\infty}g(x)=-\infty\) and \(\lim_{x
ightarrow\infty}g(x)=-\infty\).

Step6: Analyze function E: \(f(x)=x^3 + 4x^2-3\)

The leading term is \(x^3\) (degree \(n = 3\), odd; leading coefficient \(a_n = 1>0\)). As \(x
ightarrow-\infty\), \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and as \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}f(x)=\infty\).

Step7: Analyze function F: \(p(x)=-8x^2 - 3x + 10\)

The leading term is \(-8x^2\) (degree \(n = 2\), even; leading coefficient \(a_n=-8<0\)). So \(\lim_{x
ightarrow-\infty}p(x)=-\infty\) and \(\lim_{x
ightarrow\infty}p(x)=-\infty\).

Step8: Match graphs and equations in question 4

For \(y=x^3+bx^2+cx + d\) (odd - degree, positive leading coefficient), \(\lim_{x
ightarrow-\infty}y=-\infty\) and \(\lim_{x
ightarrow\infty}y=\infty\), which matches I.
For \(y =-\frac{1}{20}x^4+bx^2 + d\) (even - degree, negative leading coefficient), \(\lim_{x
ightarrow-\infty}y=-\infty\) and \(\lim_{x
ightarrow\infty}y=-\infty\), which matches IV.
For \(y=\frac{1}{20}x^4+bx^2 + d\) (even - degree, positive leading coefficient), \(\lim_{x
ightarrow-\infty}y=\infty\) and \(\lim_{x
ightarrow\infty}y=\infty\), which matches II.
For \(y =-\frac{1}{4}x^3+bx^2 + d\) (odd - degree, negative leading coefficient), \(\lim_{x
ightarrow-\infty}y=\infty\) and \(\lim_{x
ightarrow\infty}y=-\infty\), which matches III.

Answer:

A. As \(x\) decreases without bound: \(\lim_{x
ightarrow-\infty}g(x)=-\infty\), As \(x\) increases without bound: \(\lim_{x
ightarrow\infty}g(x)=\infty\)
B. As \(x\) decreases without bound: \(\lim_{x
ightarrow-\infty}p(x)=\infty\), As \(x\) increases without bound: \(\lim_{x
ightarrow\infty}p(x)=\infty\)
C. As \(x\) decreases without bound: \(\lim_{x
ightarrow-\infty}f(x)=\infty\), As \(x\) increases without bound: \(\lim_{x
ightarrow\infty}f(x)=-\infty\)
D. As \(x\) decreases without bound: \(\lim_{x
ightarrow-\infty}g(x)=-\infty\), As \(x\) increases without bound: \(\lim_{x
ightarrow\infty}g(x)=-\infty\)
E. As \(x\) decreases without bound: \(\lim_{x
ightarrow-\infty}f(x)=-\infty\), As \(x\) increases without bound: \(\lim_{x
ightarrow\infty}f(x)=\infty\)
F. As \(x\) decreases without bound: \(\lim_{x
ightarrow-\infty}p(x)=-\infty\), As \(x\) increases without bound: \(\lim_{x
ightarrow\infty}p(x)=-\infty\)

1. I - Equation: \(x^3+bx^2+cx + d\), Graph: Graph A; II - Equation: \(\frac{1}{20}x^4+bx^2 + d\), Graph: Graph C; III - Equation: \(-\frac{1}{4}x^3+bx^2 + d\), Graph: Graph D; IV - Equation: \(-\frac{1}{20}x^4+bx^2 + d\), Graph: Graph B