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Chapter 3: Problem 67

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. $$P(x)=x^{4}+c ; \quad c=-1,0,1,2$$

Short Answer

Expert verified
Changing \(c\) shifts the graph of \(x^4\) vertically by \(c\) units; up for positive \(c\) and down for negative \(c\).

Step by step solution

01

Understanding the Polynomial Function

The given polynomial is \(P(x) = x^4 + c\). This represents a family of polynomials based on the value of \(c\). The core shape is determined by \(x^4\), a quartic function, while \(c\) controls its vertical shift.
02

Define the Values of \(c\)

We need to evaluate and graph the polynomial for \(c = -1, 0, 1, 2\). Each value of \(c\) results in a vertical translation of the graph of \(x^4\).
03

Graph for \(c = -1\)

Substituting \(c = -1\) in \(P(x)\), we get \(P(x) = x^4 - 1\). This graph shifts the base graph \(x^4\) down by 1 unit.
04

Graph for \(c = 0\)

For \(c = 0\), \(P(x) = x^4 + 0 = x^4\). This is the standard graph centered at the origin with no vertical shift.
05

Graph for \(c = 1\)

With \(c = 1\), the function becomes \(P(x) = x^4 + 1\). This graph shifts the base graph \(x^4\) upward by 1 unit.
06

Graph for \(c = 2\)

Here, \(c = 2\) gives \(P(x) = x^4 + 2\). This graph shifts the base graph \(x^4\) upward by 2 units.
07

Analyzing the Effect of \(c\)

Changing the value of \(c\) vertically translates the graph \(x^4\). Increasing \(c\) moves it upward, while decreasing \(c\) moves it downward without affecting its shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are a fundamental part of algebra and calculus. They are expressions consisting of variables and coefficients, exhibiting terms that are combined using addition, subtraction, multiplication, and non-negative integer exponents. The function we are exploring here, which is given by \(P(x) = x^4 + c\), is a clear example of a polynomial function.
  • Polynomials can have one term (known as monomials), two terms (binomials), or more terms (polynomial).
  • The degree of a polynomial is determined by the term with the highest exponent. In \(x^4 + c\), the degree is 4, indicating a quartic polynomial.
  • Coefficients are the numerical factors of the terms, and in this case, the leading coefficient is 1, as indicated by \(x^4\).
Learning to recognize and graph polynomial functions helps in understanding their behavior, such as identifying zeros, turning points, and the overall shape of the graph. The core graph of \(x^4\) forms the basic structure, while the \(c\) value applies a specific transformation known as vertical translation.
Quartic Functions
Quartic functions are polynomial functions where the highest power of the variable is four. The general form of a quartic function is \(ax^4 + bx^3 + cx^2 + dx + e\), but in our case, it simplifies to \(x^4 + c\). Introducing \(c\) in this way influences the graph, while the primary quartic shape is given by \(x^4\).
  • Nearly symmetric around the vertical line through its vertex, the graph of a basic quartic \(x^4\) resembles a "U" shape where the arms expand slower than a parabola.
  • Its most distinct feature is having up to four real roots and a possible complex structure depending on its coefficients.
  • The steepness and width of the quartic's arms provide insight into its rate of change and extremities.
Recognizing that \(x^4\) forms a smooth curve displaying symmetry is vital for understanding transformations and manipulations made on such functions when other terms, like our \(c\), alter its equation.
Vertical Translation
This concept in mathematics refers to shifting a graph either upward or downward on the coordinate plane. When dealing with polynomial functions such as our quartic \(P(x) = x^4 + c\), the value of \(c\) plays a crucial role.
  • A positive \(c\) value shifts the entire graph upwards. For example, \(c = 1\) moves \(x^4\) up one unit.
  • Conversely, a negative \(c\) pushes the graph downward, which is seen when \(c = -1\), pulling it down one unit.
  • This vertical translation does not affect the shape or orientation, only the vertical position.
Understanding vertical translations helps in graphing, simplifying equations, and determining transformations within problems and real-world applications alike. Ultimately, these shifts made by \(c\) influence the positioning but preserve the inherent symmetry and form of the quartic graph.

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Most popular questions from this chapter

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, g(x)=1-x^{2}$$

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{3}+x^{2}}{x^{2}-4}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=4 x^{4}-21 x^{2}+5$$
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