3. describe the end - behavior for each polynomial function (use limit notation).\na. $g(x)=4x^{7}-3x^{4}+x$…

3. describe the end - behavior for each polynomial function (use limit notation).\na. $g(x)=4x^{7}-3x^{4}+x$ as $x$ decreases without bound: as $x$ increases without bound:\nb. $p(x)=7x^{4}+3x^{3}-3x - 4$ as $x$ decreases without bound: as $x$ increases without bound:\nc. $f(x)=-7x^{9}-8x^{3}+6$ as $x$ decreases without bound: as $x$ increases without bound:\nd. $g(x)=-3x^{6}+5x^{3}-2x + 6$ as $x$ decreases without bound: as $x$ increases without bound:\ne. $f(x)=x^{3}+4x^{2}-3$ as $x$ decreases without bound: as $x$ increases without bound:\nf. $p(x)=-8x^{2}-3x + 10$ as $x$ decreases without bound: as $x$ increases without bound:\n4. the graphs, equations, and limit statements for four polynomial functions are below. match each graph and equation to the corresponding limit statement.\ni. $lim_{x\rightarrow-infty}f(x)=infty$, $lim_{x\rightarrowinfty}f(x)=infty$ equation: graph: \nii. $lim_{x\rightarrow-infty}g(x)=infty$, $lim_{x\rightarrowinfty}g(x)=-infty$ equation: graph: \niii. $lim_{x\rightarrow-infty}h(x)=-infty$, $lim_{x\rightarrowinfty}h(x)=infty$ equation: graph: \niv. $lim_{x\rightarrow-infty}k(x)=-infty$, $lim_{x\rightarrowinfty}k(x)=-infty$ equation: graph: \na. $x^{3}+bx^{2}+cx + d$ \nb. $-\frac{1}{20}x^{4}+bx^{2}+d$ \nc. $\frac{1}{20}x^{4}+bx^{2}+d$ \nd. $-\frac{1}{4}x^{3}+bx^{2}+d$

3. describe the end - behavior for each polynomial function (use limit notation).\na. $g(x)=4x^{7}-3x^{4}+x$ as $x$ decreases without bound: as $x$ increases without bound:\nb. $p(x)=7x^{4}+3x^{3}-3x - 4$ as $x$ decreases without bound: as $x$ increases without bound:\nc. $f(x)=-7x^{9}-8x^{3}+6$ as $x$ decreases without bound: as $x$ increases without bound:\nd. $g(x)=-3x^{6}+5x^{3}-2x + 6$ as $x$ decreases without bound: as $x$ increases without bound:\ne. $f(x)=x^{3}+4x^{2}-3$ as $x$ decreases without bound: as $x$ increases without bound:\nf. $p(x)=-8x^{2}-3x + 10$ as $x$ decreases without bound: as $x$ increases without bound:\n4. the graphs, equations, and limit statements for four polynomial functions are below. match each graph and equation to the corresponding limit statement.\ni. $lim_{x\rightarrow-infty}f(x)=infty$, $lim_{x\rightarrowinfty}f(x)=infty$ equation: graph: \nii. $lim_{x\rightarrow-infty}g(x)=infty$, $lim_{x\rightarrowinfty}g(x)=-infty$ equation: graph: \niii. $lim_{x\rightarrow-infty}h(x)=-infty$, $lim_{x\rightarrowinfty}h(x)=infty$ equation: graph: \niv. $lim_{x\rightarrow-infty}k(x)=-infty$, $lim_{x\rightarrowinfty}k(x)=-infty$ equation: graph: \na. $x^{3}+bx^{2}+cx + d$ \nb. $-\frac{1}{20}x^{4}+bx^{2}+d$ \nc. $\frac{1}{20}x^{4}+bx^{2}+d$ \nd. $-\frac{1}{4}x^{3}+bx^{2}+d$

Answer

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\rightarrow\pm\infty}f(x)=\infty). If (n) is even and (a_n < 0), (\lim_{x\rightarrow\pm\infty}f(x)=-\infty). If (n) is odd and (a_n>0), (\lim_{x\rightarrow\infty}f(x)=\infty) and (\lim_{x\rightarrow-\infty}f(x)=-\infty). If (n) is odd and (a_n < 0), (\lim_{x\rightarrow\infty}f(x)=-\infty) and (\lim_{x\rightarrow-\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\rightarrow-\infty), (\lim_{x\rightarrow-\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\rightarrow\infty), (\lim_{x\rightarrow\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\rightarrow-\infty}p(x)=\infty) and (\lim_{x\rightarrow\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\rightarrow-\infty), (\lim_{x\rightarrow-\infty}f(x)=\infty) and as (x\rightarrow\infty), (\lim_{x\rightarrow\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\rightarrow-\infty}g(x)=-\infty) and (\lim_{x\rightarrow\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\rightarrow-\infty), (\lim_{x\rightarrow-\infty}f(x)=-\infty) and as (x\rightarrow\infty), (\lim_{x\rightarrow\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\rightarrow-\infty}p(x)=-\infty) and (\lim_{x\rightarrow\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\rightarrow-\infty}y=-\infty) and (\lim_{x\rightarrow\infty}y=\infty), which matches I. For (y =-\frac{1}{20}x^4+bx^2 + d) (even - degree, negative leading coefficient), (\lim_{x\rightarrow-\infty}y=-\infty) and (\lim_{x\rightarrow\infty}y=-\infty), which matches IV. For (y=\frac{1}{20}x^4+bx^2 + d) (even - degree, positive leading coefficient), (\lim_{x\rightarrow-\infty}y=\infty) and (\lim_{x\rightarrow\infty}y=\infty), which matches II. For (y =-\frac{1}{4}x^3+bx^2 + d) (odd - degree, negative leading coefficient), (\lim_{x\rightarrow-\infty}y=\infty) and (\lim_{x\rightarrow\infty}y=-\infty), which matches III.

Answer:

A. As (x) decreases without bound: (\lim_{x\rightarrow-\infty}g(x)=-\infty), As (x) increases without bound: (\lim_{x\rightarrow\infty}g(x)=\infty) B. As (x) decreases without bound: (\lim_{x\rightarrow-\infty}p(x)=\infty), As (x) increases without bound: (\lim_{x\rightarrow\infty}p(x)=\infty) C. As (x) decreases without bound: (\lim_{x\rightarrow-\infty}f(x)=\infty), As (x) increases without bound: (\lim_{x\rightarrow\infty}f(x)=-\infty) D. As (x) decreases without bound: (\lim_{x\rightarrow-\infty}g(x)=-\infty), As (x) increases without bound: (\lim_{x\rightarrow\infty}g(x)=-\infty) E. As (x) decreases without bound: (\lim_{x\rightarrow-\infty}f(x)=-\infty), As (x) increases without bound: (\lim_{x\rightarrow\infty}f(x)=\infty) F. As (x) decreases without bound: (\lim_{x\rightarrow-\infty}p(x)=-\infty), As (x) increases without bound: (\lim_{x\rightarrow\infty}p(x)=-\infty) 4. 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