Question

1 match the polynomial with its end behavior.
a $f(x)=-2x + 3$
1 as $x$ gets larger and larger in either the positive or negative direction, $f$ gets larger and larger in the positive direction.
b $f(x)=x^{2}-6x + 3$
2 as $x$ gets larger and larger in the positive direction, $f$ gets larger and larger in the positive direction.
as $x$ gets larger and larger in the negative direction, $f$ gets larger and larger in the negative direction.
c $f(x)=1 - x^{2}+2x^{3}$
3 as $x$ gets larger and larger in the positive direction, $f$ gets larger and larger in the negative direction.
as $x$ gets larger and larger in the negative direction, $f$ gets larger and larger in the positive direction.
d $f(x)=7 - x^{4}$
4 as $x$ gets larger and larger in either the positive or negative direction, $f$ gets larger and larger in the negative direction.
2 state the degree and end behavior of $f(x)=-x^{3}+5x^{2}+6x + 1$. explain or show your reasoning.

Explanation:

Problem 1:

Step1: Analyze Polynomial A

$f(x) = -2x + 3$ is degree 1, leading coefficient $-2<0$.
As $x\to+\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to+\infty$. This matches description 3.

Step2: Analyze Polynomial B

$f(x) = x^2 - 6x + 3$ is degree 2, leading coefficient $1>0$.
As $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to+\infty$. This matches description 1.

Step3: Analyze Polynomial C

$f(x) = 1 - x^2 + 2x^3$ is degree 3, leading coefficient $2>0$.
As $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to-\infty$. This matches description 2.

Step4: Analyze Polynomial D

$f(x) = 7 - x^4$ is degree 4, leading coefficient $-1<0$.
As $x\to+\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to-\infty$. This matches description 4.

Step1: Identify polynomial degree

The highest power of $x$ in $f(x) = -x^3 + 5x^2 + 6x + 1$ is 3, so degree = 3.

Step2: Determine end behavior

Leading term is $-x^3$, leading coefficient $-1<0$, odd degree.
As $x\to+\infty$, $-x^3\to-\infty$, so $f(x)\to-\infty$.
As $x\to-\infty$, $-x^3\to+\infty$, so $f(x)\to+\infty$.

Answer:

A → 3
B → 1
C → 2
D → 4

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Problem 2: