Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Modeling Polynomials Sketch the graph of a polynomial function that is of fourth degree, has a zero of multiplicity 2, and has a negative leading coefficient. Sketch another graph under the same conditions but with a positive leading coefficient.

Step-by-Step Solution

Answer

Sketch of negative leading coefficient polynomial yields both ends of the graph are pointing downward, touching the x-axis at zero and turning around. Positive leading coefficient polynomial yields both ends of the graph pointing upward, also touching the x-axis at zero and turning around.

Step 1: Sketch a polynomial with a negative leading coefficient

Since we're told it's a fourth degree polynomial with a negative leading coefficient, we know that both ends of the graph should be pointing downward. The function could be for instance \(f(x) = -x^4\). Start at the top left of the graph, draw a curve crossing the x-axis at the zero where it touches and turns around (due to multiplicity 2), and then continues downwards to the bottom right of the graph.

Step 2: Sketch a polynomial with a positive leading coefficient

Now, for a fourth degree polynomial with a positive leading coefficient, both ends of the graph should point upward. A model function can be \(g(x) = x^4\). Start at the bottom left of the graph, draw a curve crossing the x-axis at the zero where it touches and turns around (due to multiplicity 2), and then continues upwards to the top right of the graph. The two graphs should be mirror images of each other about the x-axis due to their opposite leading coefficients.

Key Concepts

Fourth Degree Polynomial

Understanding the characteristics of a fourth degree polynomial is essential when graphing these functions. A fourth degree polynomial, also known as a quartic polynomial, is an equation of the form \( ax^4 + bx^3 + cx^2 + dx + e \) where the coefficients \( a, b, c, d, \) and \( e \) are real numbers, and the leading coefficient \( a \) is not zero. The graph of a fourth degree polynomial can exhibit up to three turning points and can cross the x-axis at up to four points, referred to as its zeroes or roots.

The shape of the graph is determined by the sign and magnitude of the leading coefficient and the nature of its zeroes. If the leading coefficient is positive, the two ends of the graph will rise off to infinity, and conversely, if it is negative, the ends will fall off to infinity, creating a 'W' or 'M' shaped curve respectively.

• If the polynomial has no real zeroes, it will not intersect the x-axis at all.
• With one real zero, the graph will touch and turn back at that point.
• Multiple zeroes mean the graph will intersect the x-axis multiple times, possibly turning back at some or all of those points depending on their multiplicity.

Polynomial Zero Multiplicity

The zero multiplicity of a polynomial refers to the number of times a particular zero (root) is repeated. Multiplicity can have a significant effect on the graph of the polynomial. For instance,

• A zero with a multiplicity of 1, termed a simple zero, means the graph will cross the x-axis at this point.
• A zero with an even multiplicity, such as 2 or 4, will cause the graph to touch the x-axis and turn around, also known as a double or quadruple zero respectively.
• An odd multiplicity greater than 1, like 3, 5, etc., indicates that the graph will cross the x-axis but will flatten out a bit as it does so.

In the provided exercise, the polynomial has a zero of multiplicity 2. This means that at this zero, the graph will touch the x-axis and bounce off rather than crossing it. The 'bounce' effect creates a visual indicator on the graph that signals the presence of a zero with even multiplicity. This can be seen as a slight flattening or as a valley or peak, depending on the orientation, at the x-axis where the polynomial has that particular zero.

Visual Depiction of Multiplicity

In a graph, multiplicity will often make itself apparent in the shape of the curve. For a zero with multiplicity 2, the shape at the zero resembles the letter 'U' or an inverted 'U'.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient attached to the highest degree term in the polynomial. In the case of a fourth degree polynomial, it would be the coefficient of the \( x^4 \) term. The sign and value of the leading coefficient play a significant role in determining the graph's end behavior. Here are the basics:

• If the leading coefficient is positive, as the value of \( x \) becomes very large or very small, the polynomial will tend to infinity.
• If the leading coefficient is negative, as \( x \) becomes very large or very small, the polynomial will tend to negative infinity.

In other words, a positive leading coefficient will make the ends of the graph point upwards, resulting in a graph that may look like a hill or a series of hills that rise to the right and to the left. Conversely, a negative leading coefficient will make the ends of the graph point downwards, resembling a valley or a series of valleys. This effect is especially pertinent when graphing polynomials, as it immediately sets the overall 'direction' of the graph. Understanding this concept is critical when predicting the shape and direction of a polynomial function's graph before plotting specific points or zeroes.

Comparison with Exercise

In the exercise, one graph has a negative leading coefficient, and the other has a positive one. These lead to opposite end behaviors and overall graph shapes, mirroring each other across the x-axis. The distinction provided by the leading coefficient is fundamental for graph sketching and interpretation in algebra and precalculus studies.