Question

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$g(x)=6-4 x^{2}+x-3 x^{5}$$

Short answer

As \(x\) approaches positive infinity, \(g(x)\) approaches negative infinity; and as \(x\) approaches negative infinity, \(g(x)\) approaches positive infinity.

Step-by-step solution

Identify the Leading Term

The leading term in a polynomial function is the term that has the largest exponent. In the given polynomial function \(g(x) = 6 - 4x^{2} + x - 3x^5\), the leading term is \(- 3x^{5}\) as it possesses the highest degree (power of \(x\)), which in this case is 5.

Determine The Coefficient And Degree Sign

To determine the end behavior, we need to observe the sign of the coefficient and degree of the leading term. The leading term, \(-3x^{5}\), has a negative coefficient and an odd degree.

Describe the End Behavior

Since the leading term, \(-3x^{5}\), has a negative coefficient and an odd degree, we conclude that as \(x\) approaches positive infinity (\(x \rightarrow +\infty\)), the function \(g(x)\) approaches negative infinity (\(g(x) \rightarrow -\infty\)). This is due to the negative coefficient of the leading term. And as \(x\) approaches negative infinity (\(x \rightarrow -\infty\)), \(g(x)\) approaches positive infinity (\(g(x) \rightarrow +\infty\)). This is because of the odd degree of the leading term which inverts the sign particularly when \(x\) is negative. This is the so-called left-hand and right-hand behavior of the function.

Key concepts

Leading Term of a Polynomial

When we talk about polynomial functions, the leading term plays a crucial role in determining the function's behavior, especially as the inputs become very large or very small. The leading term is simply the term with the highest exponent. For example, consider the polynomial function g(x) = 6 - 4x^2 + x - 3x^5. In this expression, the leading term is -3x^5, because 5 is the highest power of x.

The coefficient of the leading term, in this case -3, also significantly impacts the graph of the polynomial. If this coefficient is positive, the right-hand side of the graph will rise indefinitely, whereas if it is negative, as in our example, the right-hand side of the graph will fall without bound. Understanding the leading term helps us predict the end behavior of a polynomial function.

Degree of a Polynomial

The degree of a polynomial is given by the highest exponent of its variable. It dictates the shape of the graph and the possible number of turning points. For instance, in g(x) = 6 - 4x^2 + x - 3x^5, the degree is 5, since that is the highest power of x.

The degree of a polynomial reveals several key characteristics. A polynomial with an odd degree, like g(x), typically has graphs that exhibit opposite behavior on each end: one tail will move towards positive infinity and the other towards negative infinity. Conversely, polynomials with even degrees have graphs with ends that move in the same direction. Additionally, a polynomial's degree gives us a clue about its maximum number of zeros, or roots, which is always at most the degree of the polynomial.

Coefficients of Polynomial Terms

In a polynomial function, coefficients are the numbers multiplying the variables raised to a power. Each coefficient plays a pivotal part in shaping the graph. For the polynomial g(x) = 6 - 4x^2 + x - 3x^5, the coefficients are 6, -4, 1, and -3 for the respective terms. The leading coefficient, belonging to the leading term, is especially important when determining end behavior as it indicates whether the polynomial will tend towards positive or negative infinity as x becomes very large or very small.

Polynomial coefficients also influence the steepness and stretch of the graph. A larger absolute value of a coefficient can make the graph steeper, while a smaller one would stretch it out. Understanding how each coefficient affects the polynomial can aid in graphing the function more accurately.

Graphing Polynomial Functions

The process of graphing polynomial functions involves plotting points and understanding the nature of the function based on its algebraic expression. There are significant features to consider such as intercepts, turning points, and, most importantly, end behavior. The end behavior is influenced by the leading term and its degree, as discussed previously.

For the polynomial function given by g(x) = 6 - 4x^2 + x - 3x^5, we would graph by starting with its leading term -3x^5. Since it is an odd-degree term with a negative coefficient, we know that as x → -∞, g(x) → +∞, and as x → +∞, g(x) → -∞. This information provides the 'bookends' for our graph. Within the bounds, we would calculate and plot key points including the y-intercept (where x=0), x-intercepts (if they can be found), and any turning points, which come from potential local maxima and minima in the function. By combining these points and considering the leading behavior, we can sketch a representative graph of the polynomial function.