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Magazine Chapter 3: Problem 77
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. $$P(x)=x^{4}-c x ; \quad c=0,1,8,27$$
Short Answer
Expert verified
As \( c \) increases, roots spread, and the graph dips more below the x-axis, showing increased tilting.
Step by step solution
01
Define the Polynomial Family
The given polynomial family is \( P(x) = x^{4} - c x \). Here, \( c \) is a constant that changes; we are asked to graph this polynomial for different \( c \) values: 0, 1, 8, and 27.
02
Analyze Base Case \( c = 0 \)
When \( c = 0 \), the polynomial simplifies to \( P(x) = x^4 \). This is a quartic function, which has a symmetric U-shape centered at the origin. It touches the x-axis at \( x = 0 \) and stays non-negative for all \( x \), as \( x^4 \geq 0 \) for all real \( x \).
03
Analyze \( c = 1 \)
For \( c = 1 \), the polynomial becomes \( P(x) = x^4 - x \). The graph has roots at \( x = 0 \) and \( x = 1 \). Compared to \( c = 0 \), the graph is slightly tilted and has a decrease around \( x = 1 \), resulting in a point lower than the vertex at \( x = 0 \). It goes below the x-axis between these points.
04
Analyze \( c = 8 \)
With \( c = 8 \), the polynomial is \( P(x) = x^4 - 8x \). This graph is further deformed compared to \( c = 1 \), with significant tilting. The roots are extended further apart, creating a wider section below the x-axis, showcasing increased tilting and complexity around the root points.
05
Analyze \( c = 27 \)
When \( c = 27 \), the polynomial becomes \( P(x) = x^4 - 27x \). Here, the tilting effect is even more notable. The function's roots encompass a significant range, causing a larger portion of the graph to appear below the x-axis as the linear component becomes more dominant over the quartic part.
06
Graphical Interpretation of \( c \) Changes
As \( c \) increases, the effect of the linear term \(-cx\) becomes more pronounced on the graph. The roots spread further apart, and the middle section of the graph dips more below the x-axis, indicating increased asymmetry and influence of the linear component.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
Polynomial functions express a sum of powers of an independent variable multiplied by coefficients. These versatile functions are foundational in algebra and calculus.
They can represent a wide range of curves, from simple lines and parabolas to more complex shapes.
Key characteristics of polynomial functions include:
\( P(x) = x^4 - c x \).
This function varies with different values of \( c \), showing the dynamic nature of polynomial graphs.
They can represent a wide range of curves, from simple lines and parabolas to more complex shapes.
Key characteristics of polynomial functions include:
- Degree: The highest power of the variable in the function determines the degree of the polynomial.
- Coefficients: Numbers that multiply the variable powers. They greatly influence the shape and position of the graph.
- Roots: Solutions for the polynomial set to zero. These points are where the graph crosses or touches the x-axis.
\( P(x) = x^4 - c x \).
This function varies with different values of \( c \), showing the dynamic nature of polynomial graphs.
Quartic Function
A quartic function is a polynomial of degree four. It has the general form \( ax^4 + bx^3 + cx^2 + dx + e \), where \( a \) cannot be zero.
Quartic functions can have a variety of shapes and behaviors, often exhibiting rich complexity.
In our specific exercise, considering \( P(x) = x^4 - cx \), the quartic characteristic is mainly defined by the \( x^4 \) term. Here are some features:
Quartic functions can have a variety of shapes and behaviors, often exhibiting rich complexity.
In our specific exercise, considering \( P(x) = x^4 - cx \), the quartic characteristic is mainly defined by the \( x^4 \) term. Here are some features:
- Symmetry: When there is no linear or odd-power term, the graph has perfect symmetry around the y-axis.
- Intercepts: The graph can intercept the x-axis at several points, determined by solving the equation for roots.
- Behavior: As \( x \rightarrow \pm \infty \), the term \( x^4 \) dominates, making the branches of the graph resemble a U-shape, unless altered by lower degree terms.
Effects of Constants on Graphs
Constants in polynomial expressions can dramatically change the appearance of a graph. When analyzing \( P(x) = x^4 - cx \), the constant \( c \) affects the graph in several notable ways:
- Root Spacing: Altering \( c \) shifts the roots apart or pulls them closer, changing where the graph crosses the x-axis.
- Tilt Influence: The linear term \(-cx\) introduces a tilt, decreasing symmetry, and pulling portions of the graph below the x-axis.
- Graph Asymmetry: As \( c \) increases, the influence of the linear term becomes more pronounced, significantly altering the graph鈥檚 symmetry and height.
- At \( c = 0 \), the graph is symmetric and sits entirely above the x-axis.
- With increasing \( c \), roots spread wider, the graph tilts, and a larger portion dips below the x-axis.
