Question

Find, by hand, all of the terms in each probability distribuisn, and graph. $$y=3 x^{4}-x^{3}+5 x$$

Short answer

The terms in the polynomial are \(3x^4\text{, } -x^3\text{, and } 5x\). For graphing, plots points using the polynomial for various values of x and connect them, observing the end behavior and any extrema.

Step-by-step solution

Identify Terms of the Polynomial

The first step is to identify each term of the polynomial. The given polynomial is a cubic equation of the form \(y = ax^4 + bx^3 + cx + d\), where the terms are \(3x^4\text{, } -x^3\text{, and } 5x\).

Calculate Probabilities for Each Term

Since we are asked to find the terms in each probability distribution by hand, we need to clarify that finding exact probabilities is not applicable here because we are dealing with a continuous function, not a discrete probability distribution. However, we can discuss the end behavior and relative probabilities of occurrence of the values of the function by examining the coefficients and the powers of x.

Graph the Polynomial

To graph the polynomial, plot several points for different values of \(x\) by substituting into the equation \(y = 3x^4 - x^3 + 5x\), and connect the points in a smooth curve. Note the end behavior: the polynomial goes to infinity as \(x\) goes to infinity and to negative infinity as \(x\) goes to negative infinity, because the leading term \(3x^4\) dominates for large values of \(x\). There is also a local maximum or minimum where the derivative equals to zero, this would occur where \(y'=12x^3-3x^2+5=0\).

Key concepts

Cubic Equation

A cubic equation is a polynomial equation of the third degree. The general form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a \eq 0\). In our exercise, we are analyzing a polynomial equation, \(y = 3x^4 - x^3 + 5x\), which is actually of the fourth degree, often referred to as a quartic equation. Although it is not cubic, the principles of analyzing polynomial equations apply similarly.

Despite the mix-up in the terminology of the exercise, the key takeaway remains: each term in the polynomial represents a specific behavior of the graph for different values of \(x\). For the given equation, the terms are \(3x^4\), \( -x^3\), and \(5x\). When graphing such an equation, it's crucial to consider all terms to understand how the curve behaves as it represents the combined influence of these individual components.

Polynomial End Behavior

The end behavior of a polynomial function describes the directions in which the graph of the function extends as the input value \(x\) approaches positive or negative infinity. It can be determined by the leading term, which is the term with the highest power of \(x\) in the polynomial. The coefficient of this term also plays a critical role in the end behavior. For our quartic polynomial, \(3x^4\) is the leading term, and since its coefficient is positive (\(3\)), the graph will rise to infinity when \(x\) goes to both positive and negative infinity. This is because even power functions, like \(x^4\), result in positive outcomes for both positive and negative values of \(x\), and since the coefficient is positive, the graph tends to infinity in both directions. In summary:

• As \(x\) approaches positive infinity, \(y\) also approaches positive infinity.
• As \(x\) approaches negative infinity, \(y\) also approaches positive infinity.

This concept is essential for predicting the overall shape of the graph when we cannot compute all the possible values that the function can take.

Derivative of a Polynomial

The derivative of a polynomial function gives us the slope of the tangent line to the graph at any point \(x\). It is obtained by applying the power rule to each term of the polynomial. The power rule states that for any term \(ax^n\), the derivative is \(anx^{n-1}\). For our given function, \(y = 3x^4 - x^3 + 5x\), the derivative is calculated as \(y' = 12x^3 - 3x^2 + 5\).

Finding Extrema

To find where a function reaches a local maximum or minimum, we set its first derivative to zero and solve for \(x\). For the given derivative, \(y' = 12x^3 - 3x^2 + 5 = 0\), solving this equation would give us the critical points, which may correspond to peaks (local maximums) or valleys (local minimums) on the graph. Considering a higher order derivative, such as the second derivative, would also tell us whether these critical points are indeed maxima or minima (the second derivative test).

Understanding how to compute and interpret derivatives is key to analyzing the behavior of polynomial functions, making it a powerful tool in graphing and predicting function behavior.