Problem 65 - College Algebra

Question

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or ?1. There may be more than one correct answer. The \(y\) -intercept is \((0,1)\) . There is no \(x\) -intercept. Degree is \(4 .\) End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow \infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty.\)

Step-by-Step Solution

Answer

The polynomial function could be \( f(x) = x^4 + 1 \).

Step 1: Analyze the intercepts

The given polynomial has a y-intercept at \((0,1)\) and no x-intercepts. This means the function does not cross the x-axis, suggesting it is always positive.

Step 2: Utilize the degree of the polynomial

The degree of the polynomial is 4, indicating that it is a quartic polynomial, which typically has the form \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \).

Step 3: Consider the leading coefficient and end behavior

The end behavior is described as \( x \to -\infty, f(x) \to \infty \) and \( x \to \infty, f(x) \to \infty \). This suggests that the leading coefficient \( a \) must be positive because an even-degree polynomial with a positive leading coefficient behaves this way.

Step 4: Construct the polynomial with known intercept

Since the y-intercept is \( (0,1) \), we can set \( e = 1 \) as \( f(0) \) must equal 1. Given that there are no x-intercepts, the polynomial must be expressed in such a way that it never crosses the x-axis, but is continuously positive.

Step 5: Suggest the simplest polynomial function

A potential polynomial of degree 4 that satisfies these conditions is \( f(x) = x^4 + 1 \). This polynomial has a leading positive coefficient, a y-intercept at (0, 1), and no x-intercepts since it is always 1 unit above the x-axis.

Key Concepts

Quartic Polynomials

A quartic polynomial is a polynomial of degree four. This means the highest power of the variable is four. Typically, a quartic polynomial can be written as \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \). With five coefficients (\(a, b, c, d,\) and \(e\)), quartic polynomials can create a diverse range of graph shapes and behaviors.
In the problem we're discussing, the polynomial is a quartic because the degree is specified as 4. This degree tells us that there are four roots, which might be real or complex, and that the basic shape of the graph will be quite varied.
Quartic polynomials can have anywhere from zero to four \(x\)-intercepts, but in our example, the polynomial does not touch the \(x\)-axis at all. Understanding quartic polynomials is crucial because they form the simplest set of polynomials that can exhibit all possible types of polynomial behavior.

End Behavior

End behavior of a polynomial function describes how the function behaves as \(x\) approaches positive or negative infinity. This gives us an idea of how the tails of the graph of the polynomial behave.
For a quartic polynomial, the end behavior is influenced mostly by the degree and the leading coefficient. In this exercise, regardless of which direction \( x \) approaches infinity, the function \( f(x) \) approaches infinity as well. This indicates the function’s graph opens upwards on both ends.

• An important aspect of determining end behavior is that the degree is even and the leading coefficient is positive.
• Therefore, regardless of how \( b, c, d, \) or \( e \) affect the graph's shape in different sections, the ends will always "rise".

Remember, the end behavior helps in sketching a rough graph, providing a big-picture idea of the function's shape.

Intercepts

Intercepts are points where the polynomial graph crosses the \(x\)-axis or \(y\)-axis. These points provide valuable clues about the function.
In this problem, the given polynomial has a y-intercept at \((0,1)\), meaning when \(x = 0\), the function \(f(x)\) equals 1. This is important because it helps us find the constant term \(e\) in the polynomial expression.
A curious part of this polynomial is that it lacks any \(x\)-intercepts. Simply put, this means the polynomial never crosses the \(x\)-axis. Such behavior suggests that the polynomial remains positive (or negative) for all real numbers \(x\). In our example, the polynomial is always positive, indicating no sign change and consistent direction in its course, simplifying understanding and graphing.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It provides essential insights into the graph's end behavior and direction.
In our example, the leading coefficient is 1, meaning the term \(ax^4\) has \(a = 1\).

• This positive coefficient ensures that the polynomial graph opens upwards as seen in its end behavior.
• If the leading coefficient were negative, the graph behavior would mirror, with tails descending instead.

Considering the problem, where the polynomial ends rise to infinity in both directions, the positive leading coefficient plays a critical role in shaping the graph's overall direction.
Leading coefficients not only affect the graph's broad shape but also impact whether it touches or crosses axes, aiding in visual predictability for the function.