Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of a polynomial function that satisfies the given conditions. If not possible, explain your reasoning. (There are many correct answers.) Fourth-degree polynomial with two real zeros and a negative leading coefficient

Step-by-Step Solution

Answer

A graph of such function would start in the top half of the graph, cross the x-axis at -1 and 1, and end in the lower half of the graph. The leading coefficient is negative and there are two real zeros, therefore the basic shape of the function would look like a wide 'W'.

Step 1: Formulate the Polynomial Function

By taking into account that there are two real zeros and a negative leading coefficient, a potential function can be given by the expression \(f(x) = -1*(x-a)^2*(x-b)^2\), where a and b are the real zeros.

Step 2: Plot the Zeros

The zeros a and b of the polynomial function can be plotted on the x-axis. By convention, assume a to be 1 and b to be -1. This gives \(f(x) = -1*(x-1)^2*(x+1)^2\).

Step 3: Sketch the Basic Shape of the Function

Next, using the end behavior and zeros, sketch the basic shape of the function. Since the leading coefficient is negative, the function will start from the top quadrant and end in the bottom quadrant. It will cross the x-axis at the points representing the zeros, which are -1 and 1.

Step 4: Fill in Details

To perfect your sketch, consider other features such as turning points. Since this is a fourth-degree polynomial function, there can be up to three turning points. Considering the shape, the turning points will be located between and on either side of the zeros.

Key Concepts

Fourth-Degree Polynomial

A fourth-degree polynomial is a mathematical expression of degree four, meaning the highest degree of its term is four. In general terms, it can be expressed as follows:

• \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\),

where \(a, b, c, d,\) and \(e\) are constants. One of the notable characteristics of a fourth-degree polynomial is that it can have up to four roots or zeros. Additionally, it allows for up to three turning points, where the graph changes direction.

Fourth-degree polynomials are also known as quartic polynomials. They can take on various forms depending on the specific coefficients involved, making them versatile and applicable to many real-world scenarios. In essence, these polynomials have complex yet fascinating behaviors that intertwine the concepts of zeros, turning points, and curvature changes.

Real Zeros

Real zeros of a polynomial function are the x-values for which the function equals zero. These points are located on the x-axis of the graph. For a given polynomial function\(f(x)\), the real zeros occur when \(f(x) = 0\).

In the context of a fourth-degree polynomial, there can be up to four real zeros. However, the number of real zeros depends on the graph's equation and can range from zero to four real zeros.

• The zeros determine where the graph intersects the x-axis.
• They also influence the shape and direction of the graph around those intersection points.

Finding the real zeros is crucial when sketching the graph, as these zeros give insight into the function's behavior and the structure of its graph. In exercises, real zeros are often designated by specific values, like \(a\) and \(b\), as seen in the example expression \(-1*(x-a)^2*(x-b)^2\).

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. In the expression \(ax^4\), \(a\) is the leading coefficient. This coefficient plays a significant role in determining the end behavior of the polynomial's graph.

• If the leading coefficient is positive, the ends of the graph will rise as you move from the leftmost to the rightmost ends.
• If the leading coefficient is negative, the ends of the graph will fall as you progress from left to right.

For example, if we examine a polynomial \(-1*(x-1)^2*(x+1)^2\), the leading coefficient is \(-1\). This negative leading coefficient means that the graph will have an inverted parabolic end behavior, resembling a \/ shape.

End Behavior

End behavior of a polynomial function describes what happens to the graph as the value of \(x\) approaches positive or negative infinity.

End behavior is heavily influenced by the degree of the polynomial and the sign of the leading coefficient. For a fourth-degree polynomial:

• With a positive leading coefficient, the ends will both rise.
• With a negative leading coefficient, the ends will both fall.

This helps us understand how the polynomial behaves as it moves beyond the plotted real zeros and moves towards extreme values on the x-axis.

In our case, since the polynomial function has a negative leading coefficient, the graph will start high and then dip to lower values beyond the zeros. This results in a distinct shape: beginning at the upper quadrants and descending as it reaches the outer limits of the x-axis.