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Chapter 8: Problem 10

Estimate the maximum number of turning points for each of the polynomial functions. If available, use technology to graph the function to verify the actual number. a. \(y=x^{4}-2 x^{2}-5\) c. \(y=x^{3}-3 x^{2}+4\) b. \(y=4 t^{6}+t^{2}\) d. \(y=5+x\)

Short Answer

Expert verified
a: 3, b: 5, c: 2, d: 0

Step by step solution

01

Understand the problem

Determine the maximum number of turning points for each polynomial function listed. The turning points of a polynomial occur where the first derivative changes sign.
02

Recall the rule for turning points

A polynomial function of degree n can have at most n-1 turning points.
03

Find the degree of each polynomial

Identify the highest degree term in each polynomial to find its degree.
04

Calculate the maximum number of turning points

Subtract 1 from the degree of each polynomial to find the maximum number of turning points.
05

Apply the rule to each polynomial

Evaluate each polynomial given in the problem.
06

For part (a):

The given polynomial is \( y = x^4 - 2x^2 - 5 \). The degree is 4. Maximum number of turning points = 4 - 1 = 3.
07

For part (b):

The given polynomial is \( y = 4t^6 + t^2 \). The degree is 6. Maximum number of turning points = 6 - 1 = 5.
08

For part (c):

The given polynomial is \( y = x^3 - 3x^2 + 4 \). The degree is 3. Maximum number of turning points = 3 - 1 = 2.
09

For part (d):

The given polynomial is \( y = 5 + x \). This is a linear equation with degree 1. Maximum number of turning points = 1 - 1 = 0.
10

Verification (optional)

Use graphing technology to plot each polynomial and visually confirm the number of turning points.

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Key Concepts

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

Degree of Polynomial
Polynomials can be categorized by their degree, which is the highest exponent of the variable in the polynomial. For instance, in the polynomial \( y = x^4 - 2x^2 - 5 \), the highest exponent of the variable \( x \) is 4. Therefore, the degree of this polynomial is 4. Understanding the degree of a polynomial is crucial because it directly impacts the polynomial's behavior, such as the number of turning points and the overall shape of its graph.
First Derivative
The first derivative of a polynomial function helps us understand where the function's slope changes direction, indicating turning points. Consider the polynomial \( y = x^3 - 3x^2 + 4 \). To find its first derivative, we differentiate each term with respect to \( x \):
  • \( \frac{d}{dx}(x^3) = 3x^2 \)
  • \( \frac{d}{dx}(-3x^2) = -6x \)
  • \( \frac{d}{dx}(4) = 0 \)
Combining these, the first derivative is \( y' = 3x^2 - 6x \). We then solve for where \( y' = 0 \), which gives us the potential turning points. The first derivative changing sign around these points indicates an actual turning point in the graph.
Maximum Turning Points
Turning points are where a polynomial changes direction from increasing to decreasing, or vice versa. The maximum number of turning points a polynomial can have is always one less than its degree. For example, a polynomial of degree 4, such as \( y = x^4 - 2x^2 - 5 \), can have up to 3 turning points. This rule applies to all polynomials, making it a quick and easy way to estimate how many times the graph will turn.
Graphing Technology
To visually confirm the turning points and the overall shape of a polynomial function, graphing technology can be incredibly helpful. Using a graphing calculator or software, you can input the polynomial equation, such as \( y = 4t^6 + t^2 \), and generate its graph. This allows you to observe the exact number of turning points and how the graph aligns with the theoretical maximum determined by the degree. Graphing technology not only aids in verifying calculations but also provides a deeper visual understanding of polynomial behavior.

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Most popular questions from this chapter

(Graphing program optional.) The following function represents the relationship between time \(t\) (in seconds) and height \(h\) (in feet) for objects thrown upward on Pluto. For an initial velocity of \(20 \mathrm{ft} / \mathrm{sec}\) and an initial height above the ground of 25 feet, we get $$ h=-t^{2}+20 t+25 $$ a. Find the coordinates of the point where the graph intersects the \(h\) -axis. b. Find the coordinates of the vertex of the parabola. c. Sketch the graph. Label the axes. d. Interpret the vertex in terms of time and height. e. For what values of \(t\) does the mathematical model make sense?

a. Construct a quadratic function \(Q(t)\) with exactly one zero at \(t=-1\) and a vertical intercept at -4 . b. Is there more than one possible quadratic function for part (a)? Why or why not? c. Determine the axis of symmetry. Describe the vertex.

In Exercises \(16-22,\) show that the two functions are inverses of each other. $$ F(t)=e^{3 t} \text { and } G(t)=\ln \left(t^{1 / 3}\right) $$

Using the given functions \(J, K,\) and \(L,\) where $$ J(x)=x^{3} \quad K(x)=\log (x) \quad L(x)=\frac{1}{x} $$ a. Create the function \(M(x)=(L \circ J \circ K)(x)\). b. Describe the transformation from \(x\) to \(M(x)\).

The management of a company is negotiating with a union over salary increases for the company's employees for the next 5 years. One plan under consideration gives each worker a bonus of \(\$ 1500\) per year. The company currently employs 1025 workers and pays them an average salary of \(\$ 30,000\) a year. It also plans to increase its workforce by 20 workers a year. a. Construct a function \(C(t)\) that models the projected cost of this plan (in dollars) as a function of time \(t\) (in years). b. What will the annual cost be in 5 years?
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