unit 3: polynomial functions
3.3.b independent practice & hw
1. classify each polynomial below by degree and number of terms. then, determine the degree, leading coefficient, and end behavior of each polynomial function below.
a. $f(x) = -8x + 3x^{16} - 4$
classification:
degree: leading coefficient:
end behavior:
b. $k(x) = -10x^3 + 4x^2 - 5x + 2$
classification:
degree: leading coefficient:
end behavior:
c. $y = 2(x - 3)(x + 4)(x - 1)$
classification:
degree: leading coefficient:
end behavior:
2. determine the degree of the polynomial represented in the table.
| $x_1$ | $f(x_1)$ |
| --- | --- |
| -3 | -33 |
| -2 | -12 |
| -1 | -3 |
| 0 | 0 |
| 1 | 3 |
| 2 | 12 |
| 3 | 33 |
| 4 | 72 |
| 5 | 135 |
2. determine the end behavior of each function below. then, identify whether each function has an odd or even degree and positive or negative leading coefficient.
a. image of a polynomial graph
end behavior:
degree:
leading coefficient:
b. image of a polynomial graph
end behavior:
degree:
leading coefficient:
c. image of a polynomial graph
end behavior:
degree:
leading coefficient:
d. image of a polynomial graph
end behavior:
degree:
leading coefficient:

Question

unit 3: polynomial functions
3.3.b independent practice & hw
1. classify each polynomial below by degree and number of terms. then, determine the degree, leading coefficient, and end behavior of each polynomial function below.
a. $f(x) = -8x + 3x^{16} - 4$
classification:
degree:  leading coefficient:
end behavior:
b. $k(x) = -10x^3 + 4x^2 - 5x + 2$
classification:
degree:  leading coefficient:
end behavior:
c. $y = 2(x - 3)(x + 4)(x - 1)$
classification:
degree:  leading coefficient:
end behavior:
2. determine the degree of the polynomial represented in the table.
| $x_1$ | $f(x_1)$ |
| --- | --- |
| -3 | -33 |
| -2 | -12 |
| -1 | -3 |
| 0 | 0 |
| 1 | 3 |
| 2 | 12 |
| 3 | 33 |
| 4 | 72 |
| 5 | 135 |
2. determine the end behavior of each function below. then, identify whether each function has an odd or even degree and positive or negative leading coefficient.
a. image of a polynomial graph
end behavior:
degree:
leading coefficient:
b. image of a polynomial graph
end behavior:
degree:
leading coefficient:
c. image of a polynomial graph
end behavior:
degree:
leading coefficient:
d. image of a polynomial graph
end behavior:
degree:
leading coefficient:

Accepted Answer

### Part 1a: $ f(x) = -8x + 3x^{10} - 4 $
# Explanation:
## Step1: Classify by degree and terms
The polynomial has 3 terms ( $-8x$, $3x^{10}$, $-4$ ), so it's a trinomial. The highest power (degree) is 10 (from $3x^{10}$).
## Step2: Find leading coefficient
The term with the highest degree is $3x^{10}$, so the leading coefficient is $3$.
## Step3: Determine end behavior
For a polynomial $a_nx^n+\dots$, if $n$ is even and $a_n>0$, as $x\to\pm\infty$, $f(x)\to\infty$. Here, $n = 10$ (even) and $a_n = 3>0$, so end behavior: as $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to\infty$.
# Answer (1a):
Classification: 10th - degree trinomial
Degree: 10
Leading coefficient: $3$
End Behavior: As $x\to\pm\infty$, $f(x)\to\infty$

### Part 1b: $ k(x) = -10x^3 + 4x^2 - 5x + 2 $
# Explanation:
## Step1: Classify by degree and terms
4 terms ( $-10x^3$, $4x^2$, $-5x$, $2$ ), so a quartic (degree 3) polynomial? Wait, degree is the highest power. Highest power is 3, so cubic polynomial (3rd - degree), and 4 terms → quartic? No, number of terms: 4 terms is a quadrinomial? Wait, standard: by degree (3) and number of terms (4: polynomial with 4 terms is a "four - term polynomial", but degree is 3 (cubic).
## Step2: Degree and leading coefficient
Highest power is 3 (from $-10x^3$), leading coefficient is $-10$.
## Step3: End behavior
For $n = 3$ (odd) and $a_n=-10<0$, as $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$ (since odd degree: opposite ends, and negative leading coefficient flips the right - end behavior).
# Answer (1b):
Classification: 3rd - degree four - term polynomial (or cubic polynomial)
Degree: 3
Leading coefficient: $-10$
End Behavior: As $x\to\infty$, $k(x)\to-\infty$; as $x\to-\infty$, $k(x)\to\infty$

### Part 1c: $ y = 2(x - 3)(x + 4)(x - 1) $
# Explanation:
## Step1: Expand (or find degree)
When we multiply the linear factors, the highest power will be the sum of the powers of each linear factor. Each linear factor is degree 1, and there are 3 factors, so degree $1 + 1+1 = 3$ (cubic). Number of terms: when expanded, let's see: first multiply $(x - 3)(x + 4)=x^2 + 4x-3x - 12=x^2+x - 12$, then multiply by $(x - 1)$: $x^3+x^2 - 12x-x^2 - x + 12=x^3-13x + 12$, then multiply by 2: $2x^3-26x + 24$. So 3 terms? Wait, $2(x - 3)(x + 4)(x - 1)=2[(x^2 + x - 12)(x - 1)]=2[x^3 - x^2+x^2 - x-12x + 12]=2[x^3-13x + 12]=2x^3-26x + 24$. So 3 terms (trinomial), degree 3 (cubic).
## Step2: Leading coefficient
The leading term is $2x^3$, so leading coefficient is $2$.
## Step3: End behavior
Degree 3 (odd), leading coefficient $2>0$. So as $x\to\infty$, $y\to\infty$; as $x\to-\infty$, $y\to-\infty$ (since odd degree: opposite ends, positive leading coefficient: right end up, left end down).
# Answer (1c):
Classification: 3rd - degree trinomial (cubic trinomial)
Degree: 3
Leading coefficient: $2$
End Behavior: As $x\to\infty$, $y\to\infty$; as $x\to-\infty$, $y\to-\infty$

### Part 2 (Table for $f(x_1)$):
# Explanation:
To find the degree, we can look at the finite differences.
- First differences (between consecutive $f(x_1)$):
  From $x=-3$ to $x = - 2$: $-12-(-33)=21$
  $x=-2$ to $x=-1$: $-3-(-12)=9$
  $x=-1$ to $x = 0$: $0-(-3)=3$
  $x = 0$ to $x = 1$: $3 - 0=3$
  $x = 1$ to $x = 2$: $12 - 3=9$
  $x = 2$ to $x = 3$: $33 - 12=21$
  $x = 3$ to $x = 4$: $72 - 33=39$ Wait, no, wait the table: $x = 4$, $f(x)=72$; $x = 5$, $f(x)=135$. Wait, maybe I miscalculated. Wait, let's list $x$ and $f(x)$:

$x$: $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$, $5$
$f(x)$: $-33$, $-12$, $-3$, $0$, $3$, $12$, $33$, $72$, $135$

First differences ($\Delta f$):
$-12-(-33)=21$
$-3-(-12)=9$
$0-(-3)=3$
$3 - 0=3$
$12 - 3=9$
$33 - 12=21$
$72 - 33=39$ → Wait, no, that's not symmetric. Wait, maybe second differences:

First differences: $21,9,3,3,9,21,39,63$ (wait $72 - 33 = 39$, $135 - 72=63$)

Second differences ($\Delta^2 f$):
$9 - 21=-12$
$3 - 9=-6$
$3 - 3=0$
$9 - 3=6$
$21 - 9=12$
$39 - 21=18$ → No, that's not symmetric. Wait, maybe the function is $f(x)=x^3 + 2x$? Let's test $x=-3$: $(-3)^3+2(-3)=-27 - 6=-33$ (matches). $x=-2$: $(-2)^3+2(-2)=-8 - 4=-12$ (matches). $x=-1$: $(-1)^3+2(-1)=-1 - 2=-3$ (matches). $x = 0$: $0 + 0=0$ (matches). $x = 1$: $1+2 = 3$ (matches). $x = 2$: $8 + 4=12$ (matches). $x = 3$: $27+6 = 33$ (matches). $x = 4$: $64 + 8=72$ (matches). $x = 5$: $125+10 = 135$ (matches). So $f(x)=x^3+2x$, which is a 3rd - degree polynomial.
# Answer (Part 2):
Degree: 3

### Part 2a (Graph a: symmetric about vertical line, both ends up)
# Explanation:
- End Behavior: Both ends (as $x\to\pm\infty$) go to $\infty$.
- Degree: Since both ends go in the same direction, degree is even.
- Leading Coefficient: Positive (since ends go up, and even degree with positive leading coefficient gives this end behavior).
# Answer (2a):
End Behavior: As $x\to\pm\infty$, $f(x)\to\infty$
Degree: Even
Leading Coefficient: Positive

### Part 2b (Graph b: left end up, right end down)
# Explanation:
- End Behavior: As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$.
- Degree: Opposite ends, so degree is odd.
- Leading Coefficient: Negative (since right end is down, and odd degree with negative leading coefficient gives this end behavior).
# Answer (2b):
End Behavior: As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$
Degree: Odd
Leading Coefficient: Negative

### Part 2c (Graph c: both ends down)
# Explanation:
- End Behavior: As $x\to\pm\infty$, $f(x)\to-\infty$.
- Degree: Same direction ends, so degree is even.
- Leading Coefficient: Negative (since ends go down, even degree with negative leading coefficient gives this end behavior).
# Answer (2c):
End Behavior: As $x\to\pm\infty$, $f(x)\to-\infty$
Degree: Even
Leading Coefficient: Negative

### Part 2d (Graph d: left end down, right end up)
# Explanation:
- End Behavior: As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to-\infty$.
- Degree: Opposite ends, so degree is odd.
- Leading Coefficient: Positive (since right end is up, odd degree with positive leading coefficient gives this end behavior).
# Answer (2d):
End Behavior: As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to-\infty$
Degree: Odd
Leading Coefficient: Positive

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part 1a: \( f(x) = -8x + 3x^{10} - 4 \)

Step1: Classify by degree and terms

Step2: Find leading coefficient

Step3: Determine end behavior

Step1: Classify by degree and terms

Step2: Degree and leading coefficient

Step3: End behavior

Step1: Expand (or find degree)

Step2: Leading coefficient

Step3: End behavior

Answer:

Part 1b: \( k(x) = -10x^3 + 4x^2 - 5x + 2 \)