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Chapter 4: Problem 27

(a) Determine the \(x\) - and \(y\) -intercepts and the excluded regions for the graph of the given function. Specify your results using a sketch similar to Figure \(16(a) .\) In Exercises \(31-34\) you will first need to factor the polynomial. (b) Graph each function. $$y=(x-2)(x-1)(x+1)$$

Short Answer

Expert verified
x-intercepts: -1, 1, 2; y-intercept: 2; no excluded regions; cubic graph.

Step by step solution

01

Find the x-intercepts

To find the \(x\)-intercepts, set \(y = 0\) in the equation \(y = (x-2)(x-1)(x+1)\). This occurs when any of the factors are equal to zero: \((x-2) = 0\), \((x-1) = 0\), or \((x+1) = 0\). Solving these, we find the \(x\)-intercepts are at \(x = 2\), \(x = 1\), and \(x = -1\).
02

Determine the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the function. \(y = (0-2)(0-1)(0+1) = (-2)(-1)(1) = 2\). Thus, the \(y\)-intercept is \(y = 2\).
03

Define the excluded regions

Excluded regions are typically vertical asymptotes or undefined regions. Since the given polynomial is a continuous function with no division or square roots, there are no excluded regions.
04

Sketch the graph with intercepts

To sketch the graph, plot the \(x\)-intercepts at \(x = -1\), \(x = 1\), and \(x = 2\), and the \(y\)-intercept at \(y = 2\). Connect these points, noting that the graph will have a cubic shape, starting from bottom left, going through the points and ending at top right as the leading coefficient is positive.
05

Verify polynomial shape

The polynomial \(y = (x-2)(x-1)(x+1)\) is a cubic polynomial with a standard form \(y = x^3 - 2x^2 - x + 2\). This confirms that the graph will have the typical cubic shape, passing through identified intercepts. No excluded regions are present.

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Key Concepts

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

Understanding x-intercepts in Polynomial Graphing
In the context of polynomial graphing, finding the x-intercepts is crucial. The x-intercepts are the points where the graph of the polynomial intersects the x-axis. To determine these, we need to set the function equal to zero and solve for x.
For the function \( y = (x-2)(x-1)(x+1) \), solving \( y = 0 \) involves setting each factor to zero:
  • \( x - 2 = 0 \): This gives \( x = 2 \).
  • \( x - 1 = 0 \): This gives \( x = 1 \).
  • \( x + 1 = 0 \): This gives \( x = -1 \).
Thus, the x-intercepts for this cubic polynomial are at \( x = -1, 1, \text{and } 2 \). These intercepts help in plotting the graph accurately and provide insights into the function's behavior at these points.
Determining y-intercepts in Polynomial Graphs
The y-intercept represents the point where the graph crosses the y-axis. In mathematical terms, it occurs when the value of \( x \) is zero. For our function \( y = (x-2)(x-1)(x+1) \), we find the y-intercept by substituting \( x = 0 \):
  • Calculate \( y = (0 - 2)(0 - 1)(0 + 1) \)
  • This simplifies to \( y = (-2)(-1)(1) = 2 \)
Therefore, the y-intercept is at \( y = 2 \).
This information is crucial as it provides another point on the graph and offers a starting point for plotting the polynomial.
Characteristics of Cubic Polynomials
Cubic polynomials form a fundamental category in polynomial equations, characterized by their degree of three. A typical cubic polynomial has the form \( ax^3 + bx^2 + cx + d \). For the polynomial \( y = (x-2)(x-1)(x+1) \), the expanded form is \( y = x^3 - 2x^2 - x + 2 \).
Key features of cubic polynomials include:
  • They can have up to three x-intercepts.
  • Their graphs are continuous curves.
  • Typically display an "S" shape pattern as they pass through the x-axis.
Since the leading coefficient (\( a = 1 \) in this case) is positive, the graph starts from the bottom left and exits at the top right.
This indicates the direction of the curve and assists in sketching an accurate graph.
Essentials of Graphing Polynomial Functions
Graphing functions, especially polynomials, involves identifying key characteristics such as intercepts and general shape. Knowing how to graph a function can provide a visual understanding of the function's behavior over its domain.
To effectively graph a cubic polynomial:
  • Mark the x-intercepts found from setting the polynomial to zero.
  • Locate the y-intercept, calculating where \( x = 0 \).
  • Sketch the curve, starting from the bottom and following the cubic curve's general shape.
For our example function, consistently note the curve flows smoothly through the intercepts, forming a characteristic cubic graph.
This approach helps to simplify the task of visualizing the polynomial's behavior, offering insights into changes and trends within the graph.

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Most popular questions from this chapter

An equilateral triangle of side \(x\) is inscribed in a circle. Express the area of the circle as a function of \(x\).

Sketch the graph of each rational function. Specify the intercepts and the asymptotes. (a) \(f(x)=(x-1)(x+2.75) /[(x+1)(x+3)]\) (b) \(g(x)=(x-1)(x+3.25) /[(x+1)(x+3)]\) [Compare the graphs you obtain in parts (a) and (b). Notice how a relatively small change in one of the constants can radically alter the graph.]

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Suppose that \(a\) and \(b\) are positive numbers whose sum is 1. (a) Find the maximum possible value of the product ab. (b) Prove that \(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right) \geq 9\)

You are asked to express one variable as a function of another. Be sure to state a domain for the function that reflects the constraints of the problem. The volume \(V\) and the surface area \(S\) of a sphere of radius \(r\) are given by the formulas \(V=\frac{4}{3} \pi r^{3}\) and \(S=4 \pi r^{2} .\) Express \(V\) as a function of \(S\).
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