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:
- As \(x\) approaches \(+\infty\) or \(-\infty\), \(f(x)\) will approach \(+\infty\).
- 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.