Question

directions: determine the end - behavior for the following polynomials.

1. $f(x)=-4x^{3}$

left: up
right: down

1. $g(x)=3x^{6}$

left: up
right: up

1. $y = 3(x - 1)^{3}$

left: down
right: up

1. $h(x)=8 - 3x^{4}$

left: down
right: down

1. $k(x)=8x^{2}+4 - x^{5}$

left: down
right: up

1. $m(x)=2x(x - 1)(x + 6)$

left: down
right: up

1. $p(x)=-2x(x - 3)^{2}$

left: up
right: down

1. the graphs, equations, and limit statements for four polynomial functions are below. match the graphs and equations with the correct limit statements.

limit statements
i. $lim_{x
ightarrow-infty}f(x)=-infty$, $lim_{x
ightarrowinfty}f(x)=-infty$
ii. $lim_{x
ightarrow-infty}g(x)=-infty$, $lim_{x
ightarrowinfty}g(x)=infty$
iii. $lim_{x
ightarrow-infty}h(x)=infty$, $lim_{x
ightarrowinfty}h(x)=-infty$
iv. $lim_{x
ightarrow-infty}k(x)=infty$, $lim_{x
ightarrowinfty}k(x)=infty$
function equations

1. $y=x^{3}+bx^{2}+cx + d$
2. $y=-\frac{1}{4}x^{3}+bx^{2}+d$
3. $y=-\frac{1}{20}x^{4}+cx + d$
4. $y=\frac{1}{20}x^{4}+bx^{2}+d$

graphs
graph a, graph b, graph c, graph d
limit statement: i
equation:
graph:
limit statement: ii
equation:
graph:
limit statement: iii
equation:
graph:
limit statement: iv
equation:
graph:

Explanation:

Step1: Recall end - behavior rules

The end - behavior of a polynomial \(y = a_nx^n+\cdots+a_0\) is determined by the leading term \(a_nx^n\). If \(n\) is odd and \(a_n>0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\). If \(n\) is odd and \(a_n < 0\), as \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to-\infty\). If \(n\) is even and \(a_n>0\), as \(x\to\pm\infty\), \(y\to+\infty\). If \(n\) is even and \(a_n < 0\), as \(x\to\pm\infty\), \(y\to-\infty\).

Step2: Analyze Equation 1 (\(y=x^3 + bx^2+cx + d\))

The leading term is \(x^3\) with \(a_n = 1>0\). As \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\). This matches Limit Statement II.

Step3: Analyze Equation 2 (\(y =-\frac{1}{4}x^3+bx^2 + d\))

The leading term is \(-\frac{1}{4}x^3\) with \(a_n=-\frac{1}{4}<0\). As \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to-\infty\). This matches Limit Statement I.

Step4: Analyze Equation 3 (\(y=-\frac{1}{20}x^4+cx + d\))

The leading term is \(-\frac{1}{20}x^4\) with \(a_n =-\frac{1}{20}<0\). As \(x\to\pm\infty\), \(y\to-\infty\). This matches Limit Statement III.

Step5: Analyze Equation 4 (\(y=\frac{1}{20}x^4+bx^2 + d\))

The leading term is \(\frac{1}{20}x^4\) with \(a_n=\frac{1}{20}>0\). As \(x\to\pm\infty\), \(y\to+\infty\). This matches Limit Statement IV.

Step6: Match with graphs

For a cubic function \(y =-\frac{1}{4}x^3+bx^2 + d\) (Limit Statement I, Equation 2), the graph has left - end up and right - end down, which is Graph B.
For a cubic function \(y=x^3 + bx^2+cx + d\) (Limit Statement II, Equation 1), the graph has left - end down and right - end up, which is Graph A.
For a quartic function \(y=-\frac{1}{20}x^4+cx + d\) (Limit Statement III, Equation 3), the graph has both ends down, which is Graph C.
For a quartic function \(y=\frac{1}{20}x^4+bx^2 + d\) (Limit Statement IV, Equation 4), the graph has both ends up, which is Graph D.

Answer:

Limit Statement: I
Equation: \(y =-\frac{1}{4}x^3+bx^2 + d\)
Graph: B

Limit Statement: II
Equation: \(y=x^3 + bx^2+cx + d\)
Graph: A

Limit Statement: III
Equation: \(y=-\frac{1}{20}x^4+cx + d\)
Graph: C

Limit Statement: IV
Equation: \(y=\frac{1}{20}x^4+bx^2 + d\)
Graph: D