Problem 73 - Algebra and Trigonometry

Question

\(69-74\) . Graph the family of polynomials in the same viewing rectangle, using the given values of \(c\) . Explain how changing the value of \(c\) affects the graph. $$ P(x)=x^{4}-c x, \quad c=0,1,8,27 $$

Step-by-Step Solution

Answer

As \(c\) increases, the graph skews more, shifts, and intercept distances increase.

Step 1: Identify the Polynomial Function

The polynomial given is \(P(x) = x^4 - cx\). In this function, \(c\) represents a constant that affects the graph's shape and shifted characteristics.

Step 2: Plot the Family of Polynomials

For each value of \(c\) (0, 1, 8, 27), graph the function \(P(x) = x^4 - cx\). You will plot four different graphs, one for each value of \(c\), in the same viewing window.

Step 3: Analyze the Graph when \(c=0\)

When \(c = 0\), the function simplifies to \(P(x) = x^4\). This is a standard quartic function, symmetric about the y-axis, with a minimum at the origin (0,0).

Step 4: Analyze the Graph when \(c=1\)

With \(c = 1\), the polynomial becomes \(P(x) = x^4 - x\). The graph still has similar characteristics to \(P(x) = x^4\) but will be affected by the term \(-x\), creating a slight skew; it will intersect the x-axis at 0 and at 1.

Step 5: Analyze the Graph when \(c=8\)

At \(c=8\), the polynomial is \(P(x) = x^4 - 8x\). The term \(-8x\) significantly influences the graph, resulting in shifts and potential changes in local extrema. Intersections move further from the origin.

Step 6: Analyze the Graph when \(c=27\)

For \(c=27\), the polynomial becomes \(P(x) = x^4 - 27x\). The term \(-27x\) causes a marked shift in the graph, creating intersections far from the origin, signaling crushed valleys and exaggerated extrema as compared to \(c=8\).

Step 7: Summary of Effect of c on Graph

As \(c\) increases, the linear term \(-cx\) dominates more, increasing the graph's skew and shifting intercepts. The overall shape flattens between roots, and vertical extent grows. The quartic behavior is overpowered by the linear term at larger \(c\) values.

Key Concepts

Quartic Functions

A quartic function is a type of polynomial function where the highest power of the variable is 4. In the equation \(P(x) = x^4 - cx\), the \(x^4\) term is what makes it a quartic function. These functions are known for having a characteristic shape that resembles a W or M, depending on the coefficients and constant terms involved.
Quartic functions can have up to four real roots, but the number of turning points, which are the peaks and troughs of the graph, is always three or fewer. This is because the derivative of a quartic function is a cubic polynomial, which can have at most three real roots.
The basic form of a quartic function with no additional constant or linear terms, \(P(x) = x^4\), results in a symmetric graph centered on the y-axis, with a single global minimum at the origin \((0,0)\). This symmetry is broken when additional terms like \(-cx\) are introduced.

Polynomial Characteristics

Polynomials have several key characteristics that define their graph and behavior. Some of these characteristics include:

• Degree: This refers to the highest power of \(x\) in the polynomial. Higher-degree polynomials like quartics (degree 4) tend to have more complex graphs.
• End Behavior: For quartics, since the leading term \(x^4\) dominates, the ends of the graph will both rise to infinity, resembling the behavior of even-degree polynomials.
• Turning Points: As mentioned, quartic functions can have up to three turning points. These are where the graph changes direction from increasing to decreasing or vice versa.
• Roots: The roots are the points where the polynomial crosses the x-axis. These are also known as zeros or solutions.

Understanding these characteristics helps in sketching and interpreting polynomial graphs effectively. By analyzing these features, you can predict how changes in constants or coefficients will affect the overall look and position of the graph.

Effect of Constants on Graphs

In the polynomial equation \(P(x) = x^4 - cx\), the constant \(c\) directly influences the overall shape and position of its graph. As \(c\) changes, it particularly affects the linear component \(-cx\) and its influence on the graph:

• When \(c = 0\), the polynomial is simply \(P(x) = x^4\). The graph is symmetric and centered, with its minimum point at the origin.
• For \(c = 1\), the graph starts to tilt slightly, changing its symmetry, and new x-intercepts emerge, such at x=1 due to the \(-cx\) term.
• As \(c\) increases to values like 8 and 27, the \(-cx\) term heavily influences the polynomial, causing the graph's turning points and intercepts to spread further away from the origin. It changes the positioning of the roots and steepens the graph between roots compared to when \(c\) is smaller.

Essentially, as \(c\) grows larger, the graph's quartic nature is somewhat overshadowed by the linear term, leading to greater shifts and distortions in both the size and shape in correlation with increasing \(c\) values. The flattening and stretching effects thus become pronounced in these cases.