Use the equation of the polynomial function ​f(x)equalsnegative left parenthesis x minus 4 right parenthesis left parenthesis x squared minus 4 right parenthesis to complete the following.

Question

​(a) State the degree and the leading coefficient.
​(b) Describe the end behavior of the graph of the function.
​(c) Support your answer by graphing the function.
Question content area bottom
Part 1
​(a) The degree of the polynomial is
  
enter your response here and the leading coefficient is
  
enter your response here.
Part 2
​(b) Describe the end behavior of the graph of the function.
The curve opens
▼
down
up
to the right because the leading coefficient is
▼
negative
positive
. Because the polynomial is
▼
cubic,
quartic,
the graph has end behaviors in the
▼
opposite
same
​direction, so the other end opens
▼
down
up
to the left.
Part 3
​(c) Which of the following is the correct graph of the function ​f(x)equalsnegative left parenthesis x minus 4 right parenthesis left parenthesis x squared minus 4 right parenthesis​?
A.
-8
8
-60
60
x
y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 60 to 60 in increments of 15. From left to right, a curve rises at a decreasing to a maximum comma then falls at an increasing then decreasing rate to a minimum comma then rises at an increasing then decreasing rate to a maximum comma and then falls at an increasing rate. The curve passes through the following points: (negative 2, 0); (negative 1, 6); (0, 0); (1, negative 3); (2, 0). All coordinates are approximate.
B.
-8
8
-60
60
x
y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 60 to 60 in increments of 15. From left to right, a curve rises at a decreasing to a maximum comma then falls at an increasing then decreasing rate to a minimum comma and then rises at an increasing rate. The curve passes through the following points: (negative 2, 0); (negative 1, 15); (0, 16); (1, 9); (2, 0). All coordinates are approximate.
C.
-8
8
-60
60
x
y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 60 to 60 in increments of 15. From left to right, a curve falls at a decreasing to a minimum comma then rises at an increasing then decreasing rate to a maximum comma and then falls at an increasing rate. The curve passes through the following points: (negative 2, 0); (negative 1, negative 15); (0, negative 16); (1, negative 9); (2, 0). All coordinates are approximate.
D.
-8
8
-60
60
x
y

Answer 1

Let's start by analyzing the polynomial function given by:

\[ f(x) = - (x - 4)(x^2 - 4) \]

Part 1: Degree and Leading Coefficient

First, we need to expand the polynomial to find its degree and leading coefficient.

1. Expand \( f(x) \): \[ f(x) = - (x - 4)(x^2 - 4) \] Distributing \( (x - 4) \): \[ = -[(x \cdot (x^2 - 4)) - (4 \cdot (x^2 - 4))] \] \[ = -[x^3 - 4x - 4x^2 + 16] \] The expansion gives: \[ f(x) = -x^3 + 4x^2 + 16x - 16 \]

2. From the polynomial \( f(x) = -x^3 + 4x^2 + 16x - 16 \), we can identify:

   • The degree is 3 (the highest exponent).
   • The leading coefficient is -1 (the coefficient of the \( x^3 \) term).

Thus,

(a) The degree of the polynomial is 3 and the leading coefficient is -1.

Part 2: End Behavior

The leading coefficient is negative, and the degree is odd (3), which affects the end behavior:

• As \( x \to \infty \) (the right end), \( f(x) \to -\infty \).
• As \( x \to -\infty \) (the left end), \( f(x) \to \infty \).

Therefore, the description of the end behavior is:

The curve opens down because the leading coefficient is negative. Because the polynomial is cubic, the graph has end behaviors in opposite directions, hence the other end opens up to the left.

(b) The curve opens down to the right because the leading coefficient is negative. Because the polynomial is cubic, the graph has end behaviors in the opposite direction, so the other end opens up to the left.

Part 3: Graph of the Function

Based on the characteristics of the polynomial we identified:

1. The function will have the following end behavior:

   • Left side goes up (as \( x \to -\infty \)).
   • Right side goes down (as \( x \to \infty \)).

2. A cubic polynomial typically has inflection points and may have up to two local extrema.

Among the provided options, we need to look for a graph that shows:

• An upward trend as you move to the left.
• A downward trend as you move to the right.

Considering the provided graphs:

• Option C portrays a downward trend towards the right and an upward trend towards the left, which aligns with the end behaviors described above.

Thus, the correct graph of the function \( f(x) \) is most likely C.