Question

Use the equation of the polynomial function ​f(x)equals4 x Superscript 4 Baseline minus x to complete the following.
​(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**.