Problem 47 - Precalculus: Functions and Graphs

Question

(a) Construct a table containing the values of the fourthdegree polynomials $$ \begin{array}{l} f(x)=2 x^{4} \\ g(x)=2 x^{4}-5 x^{2}+1 \\ h(x)=2 x^{4}+5 x^{2}-1 \end{array} $$ and $$ k(x)=2 x^{4}-x^{3}+2 x $$ when \(x=\pm 20, \pm 40,\) and \(\pm 60\) (b) As \(|x|\) becomes large, how do the values for each function compare? Which term has the greatest influence on each function's value when \(|x|\) is large?

Step-by-Step Solution

Answer

As |x| becomes large, 2x^4 dominates in each polynomial.

Step 1: Calculate f(x) for Given x Values

First, calculate the values for the polynomial \( f(x) = 2x^4 \) at \( x = \pm 20, \pm 40, \pm 60 \). Note that since this is a pure fourth-degree polynomial, we only need to compute \( 2(x^4) \). Compute for each \( x \): \[ \begin{align*} f(20) &= 2(20^4), \ f(-20) &= 2((-20)^4), \ f(40) &= 2(40^4), \ f(-40) &= 2((-40)^4), \ f(60) &= 2(60^4), \ f(-60) &= 2((-60)^4). \end{align*} \] Calculate these values: \[ f(20) = 2(160,000) = 320,000, \quad f(-20) = 320,000 \]\[ f(40) = 2(2,560,000) = 5,120,000, \quad f(-40) = 5,120,000 \]\[ f(60) = 2(12,960,000) = 25,920,000, \quad f(-60) = 25,920,000 \].

Step 2: Calculate g(x) for Given x Values

Next, calculate for \( g(x) = 2x^4 - 5x^2 + 1 \) using same \( x \) values. Note the quadratic term \(-5x^2\) and constant term \(+1\).\[ \begin{align*} g(20) &= 2(20^4) - 5(20^2) + 1, \ g(-20) &= 2((-20)^4) - 5((-20)^2) + 1, \ g(40) &= 2(40^4) - 5(40^2) + 1, \ g(-40) &= 2((-40)^4) - 5((-40)^2) + 1, \ g(60) &= 2(60^4) - 5(60^2) + 1, \ g(-60) &= 2((-60)^4) - 5((-60)^2) + 1. \end{align*} \]Calculate resulting terms:- \( g(20) = 320,000 - 2000 + 1 = 318,001 \) - \( g(40) = 5,120,000 - 8,000 + 1 = 5,112,001 \) - \( g(60) = 25,920,000 - 18,000 + 1 = 25,902,001 \). Due to symmetry, \( g(-x) = g(x) \) for each \( x \).

Step 3: Calculate h(x) for Given x Values

Now compute \( h(x) = 2x^4 + 5x^2 - 1 \). Use similar calculations:\[ \begin{align*} h(20) &= 2(20^4) + 5(20^2) - 1, \ h(-20) &= h(20), \ h(40) &= 2(40^4) + 5(40^2) - 1, \ h(-40) &= h(40), \ h(60) &= 2(60^4) + 5(60^2) - 1, \ h(-60) &= h(60). \end{align*} \] Calculate these values:- \( h(20) = 320,000 + 2000 - 1 = 321,999 \)- \( h(40) = 5,120,000 + 8,000 - 1 = 5,127,999 \)- \( h(60) = 25,920,000 + 18,000 - 1 = 25,937,999 \). Again, due to symmetry, \( h(-x) = h(x) \).

Step 4: Calculate k(x) for Given x Values

Finally, compute \( k(x) = 2x^4 - x^3 + 2x \). Calculation modifies with cubic and linear terms:\[ \begin{align*} k(20) &= 2(20^4) - (20^3) + 2(20), \ k(-20) &= 2((-20)^4) - (-20)^3 + 2(-20), \ k(40) &= 2(40^4) - (40^3) + 2(40), \ k(-40) &= 2((-40)^4) - (-40)^3 + 2(-40), \ k(60) &= 2(60^4) - (60^3) + 2(60), \ k(-60) &= 2((-60)^4) - (-60)^3 + 2(-60). \end{align*} \] Compute:- \( k(20) = 320,000 - 8000 + 40 = 312,040 \) - \( k(-20) = 320,000 + 8000 - 40 = 327,960 \) - \( k(40) = 5,120,000 - 64,000 + 80 = 5,056,080 \) - \( k(-40) = 5,120,000 + 64,000 - 80 = 5,183,920 \)- \( k(60) = 25,920,000 - 216,000 + 120 = 25,704,120 \)- \( k(-60) = 25,920,000 + 216,000 - 120 = 26,135,880 \).

Step 5: Analyze Large |x| Behavior

For large \(|x|\), the leading term \( 2x^4 \) dominates in each polynomial as it grows faster than any lower-degree term. Compare for all functions:- For \( f(x) = 2x^4 \), values scale purely on quartic growth.- For \( g(x) \) and \( h(x) \), additional terms minimally adjust close values due to dominance of \( 2x^4 \).- For \( k(x) \), while \( -x^3 \) shifts the result, leading quartic term ensures dominance at large scales.In all cases, the term \( 2x^4 \) is the most significant.

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables that are combined using addition, subtraction, and multiplication. They feature a structure where each term is a product of a constant and a variable raised to a non-negative integer power. The standard form of a polynomial is written as: \[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where \( a_n \), \( a_{n-1} \), ..., \( a_1 \), and \( a_0 \) are constants, and \( n \) represents the degree of the polynomial, which is determined by the term with the largest exponent.
In the context of the exercise, each polynomial begins with a fourth-degree term \( 2x^4 \). This makes them fourth-degree polynomials, also known as quartic polynomials. There are other terms with lower powers of x, such as \( -5x^2 \) or \( -x^3 \), including constant terms like \(+1\) and \-1\, but the term \( 2x^4 \) dictates the primary behavior, especially as values of \( x \) increase.

Leading Term Behavior

The leading term of a polynomial refers to the term with the highest power of \( x \). Its significance comes from its dominant influence on the polynomial's behavior as \( x \) becomes very large or very small. In a quartic polynomial like those in the exercise, the leading term is \( 2x^4 \). This term dictates how the function behaves at the extremes of its domain.

• As \(|x|\) becomes large, the value of \( 2x^4 \) grows much faster than any lower-degree terms, such as \(-5x^2\) or \(-x^3\).
• The rapid increase or decrease dictated by the leading term overshadows the influence of the smaller power terms as \(|x|\) increases.
• This concept illustrates why, despite having additional terms, the overall end behavior of the polynomial hinges mostly on the leading term. If you were to graph these polynomials, you'd observe their shapes align closely with the graph of \( 2x^4 \), especially for larger \(|x|\).

Understanding this helps predict the behavior of polynomial functions just by looking at the highest degree term.

Quartic Growth

Quartic growth refers to the rate at which a fourth-degree polynomial function increases or decreases. The term "quartic" comes from the Latin word "quartus," meaning fourth. A quartic polynomial like \( f(x) = 2x^4 \) will exhibit quartic growth, which is faster than quadratic or cubic growth patterns.

• When \( x \) is positive and large, \( f(x) \) becomes very large at a rate proportional to \( x^4 \).
• The function increases rapidly, much quicker than it would if it were quadratic \( x^2 \) or cubic \( x^3 \).
• Similarly, when \( x \) is negative, the quartic term maintains a positive outcome because even powers of negative numbers are positive. Thus, \( f(x) \) will also grow rapidly with increasing \(|x|\).

This growth pattern makes quartic functions particularly interesting in mathematics due to their steep climbs and significant changes in value, compared to lower degree polynomials. Understanding quartic growth helps anticipate the scale differences when dealing with polynomial functions of various degrees.