Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=\frac{3 x^{4}-2 x+5}{4}$$

Step-by-Step Solution

Answer

The right-hand and left-hand behavior of the graph of the given polynomial function both tend towards positive infinity (\(+\infty\)).

Step 1: Analyze the degree and leading coefficient of the polynomial

The degree of the polynomial is the highest power of x, in this case 4 (which is an even number). The leading coefficient is the coefficient of the highest degree term, in this case 3 (which is positive).

Step 2: Deduce the end behavior

For a polynomial, the sign of the leading coefficient and the degree of the function determine the end behavior of the graph. Because the degree is even (4) and the leading coefficient is positive (3), as x approaches positive infinity (\(x \rightarrow +\infty\)) and negative infinity (\(x \rightarrow -\infty\)), the value of the function \(f(x)\) approaches positive infinity (\(f(x) \rightarrow +\infty\)). This is a standard rule for polynomials.

Key Concepts

End Behavior

When we talk about the end behavior of a polynomial function, we refer to what happens to the function's values as the input, \(x\), becomes very large (approaches infinity) or very small (approaches negative infinity).

Understanding end behavior helps us predict how the graph of a polynomial will behave at its extremes, much like glancing at the early minutes of a movie hint towards the ending. A polynomial's end behavior is largely influenced by two aspects:

• the degree of the polynomial,
• the sign of the leading coefficient.

For instance, in the provided function \(f(x) = \frac{3x^4 - 2x + 5}{4}\), the polynomial part is dominated by the term \(3x^4\). Because the degree of this term is even (4) and the leading coefficient (3) is positive, both ends of the graph will rise upwards as \(x\) moves towards positive or negative infinity. Ultimately, this means \(f(x)\) heads to positive infinity in both directions.

Leading Coefficient

The leading coefficient is the coefficient attached to the term with the highest power of \(x\) in a polynomial. It plays a crucial role in determining how the polynomial's graph appears, especially regarding its steepness and direction of the end behavior.

If the leading coefficient is positive, as in our example \(f(x) = \frac{3x^4 - 2x + 5}{4}\), this suggests the graph will open upwards for even degree polynomials. Conversely, a negative leading coefficient, for an even degree polynomial, would flip the graph, making it open downwards.

• A positive leading coefficient:
  • Graph ascends toward \(+\infty\) as \(x\rightarrow\infty\) or \(-\infty\).
• A negative leading coefficient:
  • Graph descends, tending towards \(-\infty\).

The leading coefficient does not alter the symmetry around the center, but it dictates the vertical direction at the graph's ends.

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of the variable \(x\) that appears in the polynomial expression when simplified. This is a key concept because it provides insights into both the number of turning points and the general end behavior of the function.

In the function \(f(x) = \frac{3x^4 - 2x + 5}{4}\), the term with the highest power of \(x\) is \(3x^4\), which gives it a degree of 4. This number being even holds significance for the polynomial's symmetry and the direction the graph takes on both ends of the axis.

• For even degree:
  • Both ends of the graph tend to move in the same direction.
• For odd degree:
  • The ends of the graph will move in opposite directions, creating a departure and approach like a zig-zag.

Therefore, the degree helps predict the graph's appearance, shaping how many local maxima or minima could be present.