Problem 50 - Precalculus: Functions and Graphs

Question

(a) Graph each of the following fourth-degree polynomials \(f\) in the viewing rectangle \([-9,9]\) by \([-6,6]\) (1) \(f(x)=-x^{4}-2 x^{3}+5 x^{2}+6 x-3\) (2) \(f(x)=x^{4}-2 x^{3}+1\) (3) \(f(x)=-\frac{1}{2} x^{4}+2 x^{2}-x+1\) \(f(x)=\frac{1}{3} x^{4}-\frac{1}{2} x^{3}-\frac{7}{3} x^{2}+\frac{7}{2} x+3\) (b) Discuss the shape of the graph of \(f\) as \(|x|\) becomes large. (c) Make a generalization about the end behavior of the function \(f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e\)

Step-by-Step Solution

Answer

The end behavior of a fourth-degree polynomial is dominated by the leading term. If the leading coefficient is positive, both ends rise; if negative, both ends fall.

Step 1: Graphing the Polynomial Function

Graph the given polynomial functions one by one using a graphing utility within the viewing rectangle [-9, 9] by [-6, 6]. For graph (1), you will notice the shape is dominated by the leading term \(-x^4\). The highest power is an even degree, indicating that as \(x\to\pm \infty\), \(f(x)\to -\infty\). Graph (2) with \(x^4\) will show \(f(x)\to \infty\ as \,x\to\pm \infty\). Graph (3), with \(-\frac{1}{2}x^4\), shows \(f(x)\to -\infty\) for large \(|x|\). Graph (4) with \(x^4\) dominates as \(x\to\pm \infty\).

Step 2: Analyze the Graph Shape as |x| Becomes Large

Observe that the shape of each polynomial graph is primarily determined by the leading term. For even-degree polynomials, if the leading coefficient is positive, both ends of the graph rise upwards, and if negative, both ends fall downwards. In our examples, graphs (1) and (3) fall as \(|x|\) becomes large, while graphs (2) and (4) rise.

Step 3: Generalize the End Behavior

For a general fourth-degree polynomial \(f(x)=ax^4+bx^3+cx^2+dx+e\), the end behavior is dictated by the term \(ax^4\). If \(a>0\), as \(|x|\to\infty\), \(f(x)\to\infty\). If \(a<0\), as \(|x|\to\infty\), \(f(x)\to-\infty\), since the highest power overrides any lower-degree terms as \(|x|\) becomes very large.

Key Concepts

Graphing Polynomials

Graphing polynomials is a fundamental concept in understanding how these mathematical functions behave visually. This process involves plotting the function on a coordinate plane, which helps in identifying critical points like zeros, maxima, and minima. When graphing any polynomial, especially ones like our fourth-degree examples, start by considering the degree and leading coefficient. These elements significantly influence the overall shape of the graph.

• Use a graphing utility to accurately represent the polynomial function's behavior.
• Select an appropriate viewing window: in our examples, the rectangle egin{math} [-9, 9] imes [-6, 6] end{math} is suggested.
• Plot each function, looking out for specific features such as turning points and intercepts.

This step provides a visual reference that serves as the foundation for deeper analysis of the polynomial's properties.

End Behavior

The end behavior of a polynomial function describes how the function's values behave as the input values become very large or very small (i.e., as egin{math} |x| end{math} approaches infinity). This behavior is predominantly determined by the polynomial's leading term.

For any polynomial function, two key factors come into play to determine its end behavior:

• The degree of the polynomial: Even-degree polynomials like the fourth-degree will have similar behavior on both ends, either both up or both down.
• The sign of the leading coefficient: Positive coefficients lead to egin{math} f(x) o rac{ ext{direction}}{\underline{\phantom{xx}}} end{math} positive infinity, whereas negative coefficients lead to a direction towards negative infinity.

By examining these factors, you can predict how the function "ends"—whether in the top or bottom end of the graph—as the value of egin{math} |x| end{math} increases.

Fourth-degree Polynomials

Fourth-degree polynomials, also known as quartic polynomials, have specific characteristics due to their power. These functions are defined by the general form egin{math} ax^4 + bx^3 + cx^2 + dx + e end{math}, where egin{math} a eq 0 end{math}. This degree ensures that the polynomial will have certain sets of symmetric properties when graphed.

• Can have up to four roots or zeros, real or complex.
• Typically display three turning points, making them more complex than lower-degree polynomials like cubics.
• Exhibit an "M" or "W" shape depending on the sign and size of the leading coefficient.

These properties make fourth-degree polynomials fascinating subjects for analysis and visualization.

Leading Term Behavior

The leading term in a polynomial function is the term with the highest degree, and it plays a crucial role in determining the polynomial's shape and behavior. In our examples of fourth-degree polynomials, this is egin{math} ax^4 end{math}.

Understanding how this term influences the function allows you to make accurate predictions about the graph's behavior as egin{math} |x| end{math} becomes very large.

• If egin{math} a > 0 end{math}, the polynomial's graph will open upwards at both ends.
• If egin{math} a < 0 end{math}, it will open downwards at both ends.
• Any lower-degree terms ( egin{math} bx^3, cx^2, ext{etc.} end{math}) become less significant as egin{math} |x| end{math} increases, emphasizing the dominance of egin{math} ax^4 end{math}.

This dominance of the leading term is what fundamentally characterizes the end behavior and overall profile of a polynomial function, making it a critical component in the analysis of polynomials.