Use the equation of the polynomial function ​f(x)equals4 x Superscript 4 Baseline minus x 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
▼
positive
negative
. Because the polynomial is
▼
quartic,
cubic,
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)equals4 x Superscript 4 Baseline minus x​?
A.
-2
2
-2
2
x
y

A coordinate system has a horizontal x-axis labeled from negative 2 to 2 in increments of 1 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. From left to right, a curve falls steeply at a decreasing rate passing through the point left parenthesis negative 0.5 comma 0 right parenthesis to a minimum comma and rises steeply at an increasing rate passing through the point left parenthesis 0 comma 0 right parenthesis to a maximum comma and falls steeply at an increasing rate passing through the point left parenthesis 0.5 comma 0 right parenthesis.
B.
-2
2
-2
2
x
y

A coordinate system has a horizontal x-axis labeled from negative 2 to 2 in increments of 1 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. From left to right, a curve rises steeply at a decreasing rate passing through the point left parenthesis 0 comma 0 right parenthesis comma and rises steeply at an increasing rate.
C.
-2
2
-2
2
x
y

A coordinate system has a horizontal x-axis labeled from negative 2 to 2 in increments of 1 and a vertical y-axis labeled from negative 2 to 2 in increments of 1. From left to right, a curve rises steeply at a decreasing rate passing through the point left parenthesis negative 0.6 comma 0 right parenthesis to a maximum comma falls at an increasing rate passing through the point left parenthesis 0 comma 0 right parenthesis comma and falls steeply at an increasing rate.
D.
-2
2
-2
2
x
y

Answer 1

Let's analyze the polynomial function \( f(x) = 4x^4 - x \) step by step to complete the questions.

Part 1: Degree and Leading Coefficient

(a) The degree of the polynomial is 4 (since the highest power of \(x\) is 4) and the leading coefficient is 4 (the coefficient of the highest power \(x^4\)).

Part 2: End Behavior

(b) To describe the end behavior of the graph:

• The leading term is \(4x^4\), which has a positive leading coefficient.
• Because the polynomial is quartic (degree 4), the graph behaves the same way at both ends (i.e., both ends either rise or fall together).
• Therefore, the curve opens up to the right and also up to the left.

Part 3: Graphing the Function

(c) Now, let's analyze the given graph options.

• A quartic function with a positive leading coefficient (such as \(4x^4\)) will typically have the following features:
  1. As \(x\) approaches \(+\infty\) or \(-\infty\), \(f(x)\) will approach \(+\infty\).
  2. It typically has a minimum point where it approaches from above, indicating that there can be points where it crosses the x-axis.

Based on these properties, the correct description of the graph would be one that rises steeply on both ends and may have a minimum or inflection point in between.

This describes graph C, which mentions that it rises steeply at a decreasing rate to a maximum, falls at an increasing rate, and falls steeply at an increasing rate.

Final Answers:

• (a) The degree of the polynomial is 4 and the leading coefficient is 4.
• (b) The curve opens up to the right because the leading coefficient is positive. Because the polynomial is quartic, the graph has end behaviors in the same direction, so the other end opens up to the left.
• (c) The correct graph of the function \(f(x) = 4x^4 - x\) is C.