Question

complete the table to identify the leading coefficient, degree, and end behavior of each polynomial.

	polynomial	leading coefficient	degree	graph comparison	end behavior
8	$f(x) = -7x^9 + 2x^2 - 3x$
9	$f(x) = -2x^5 - x^3 + 6x$

1. identify whether each function graphed has an odd or even degree and a positive or negative leading coefficient.

for the first graphed function:
degree: __________
leading coefficient: ______

for the second graphed function (with grid):
degree: __________
leading coefficient: ______

for the third graphed function:
degree: __________
leading coefficient: ______

Explanation:

Part 7: Polynomial \( f(x) = 4x^7 + 5x^4 + 2 \)

Step 1: Find Leading Coefficient

The leading term is the term with the highest degree. Here, the highest degree term is \( 4x^7 \), so the leading coefficient is \( 4 \).

Step 2: Find Degree

The degree of a polynomial is the highest power of \( x \). For \( 4x^7 \), the power is \( 7 \), so the degree is \( 7 \).

Step 3: Graph Comparison

The leading term \( 4x^7 \) has an odd degree and positive leading coefficient. So its graph behaves like \( y = x^7 \) (odd, positive leading coefficient: as \( x \to +\infty \), \( y \to +\infty \); as \( x \to -\infty \), \( y \to -\infty \)).

Step 4: End Behavior

Since degree is odd (\( 7 \)) and leading coefficient is positive (\( 4 \)): as \( x \to +\infty \), \( f(x) \to +\infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

Part 8: Polynomial \( f(x) = -7x^9 + 2x^2 - 3x \)

Step 1: Find Leading Coefficient

The leading term is \( -7x^9 \), so leading coefficient is \( -7 \).

Step 2: Find Degree

Highest power of \( x \) is \( 9 \), so degree is \( 9 \).

Step 3: Graph Comparison

Leading term \( -7x^9 \) has odd degree and negative leading coefficient. Behaves like \( y = -x^9 \) (odd, negative: as \( x \to +\infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to +\infty \)).

Step 4: End Behavior

Degree is odd (\( 9 \)), leading coefficient negative (\( -7 \)): as \( x \to +\infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to +\infty \).

Part 9: Polynomial \( f(x) = -2x^5 - x^3 + 6x \)

Step 1: Find Leading Coefficient

Leading term is \( -2x^5 \), so leading coefficient is \( -2 \).

Step 2: Find Degree

Highest power of \( x \) is \( 5 \), so degree is \( 5 \).

Step 3: Graph Comparison

Leading term \( -2x^5 \) has odd degree and negative leading coefficient. Behaves like \( y = -x^5 \) (odd, negative: as \( x \to +\infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to +\infty \)).

Step 4: End Behavior

Degree is odd (\( 5 \)), leading coefficient negative (\( -2 \)): as \( x \to +\infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to +\infty \).

Part 10: Graph 1 (Leftmost)

Answer:

s for Graphs:

• Graph 1 (Left):

Degree: Odd
Leading Coefficient: Positive

• Graph 2 (Middle):

Degree: Even
Leading Coefficient: Positive

• Graph 3 (Right):

Degree: Odd
Leading Coefficient: Negative

Final Answers (Table Summary for Polynomials):

• For \( f(x) = 4x^7 + 5x^4 + 2 \): Leading Coefficient \( \boldsymbol{4} \), Degree \( \boldsymbol{7} \), End Behavior: \( x \to +\infty, f(x) \to +\infty \); \( x \to -\infty, f(x) \to -\infty \)
• For \( f(x) = -7x^9 + 2x^2 - 3x \): Leading Coefficient \( \boldsymbol{-7} \), Degree \( \boldsymbol{9} \), End Behavior: \( x \to +\infty, f(x) \to -\infty \); \( x \to -\infty, f(x) \to +\infty \)
• For \( f(x) = -2x^5 - x^3 + 6x \): Leading Coefficient \( \boldsymbol{-2} \), Degree \( \boldsymbol{5} \), End Behavior: \( x \to +\infty, f(x) \to -\infty \); \( x \to -\infty, f(x) \to +\infty \)

Final Answers (Graphs):

• Graph 1: Degree: Odd, Leading Coefficient: Positive
• Graph 2: Degree: Even, Leading Coefficient: Positive
• Graph 3: Degree: Odd, Leading Coefficient: Negative