Book cover for Algebra and Trigonometry

Algebra and Trigonometry

Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger

ISBN #9780321693983

4th Edition

5,839 Questions

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199,024 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section elaborates on the behavior of functions by categorizing them into increasing, decreasing, and constant intervals. It integrates graphical analysis with calculus concepts such as slopes to identify these regions and further explains relative maxima and minima. The content also highlights the importance of accurately determining the domain, particularly when dealing with piecewise functions, and demonstrates applications to real-world problems like distance measurement and area optimization.

Learning Objectives

1

Determine and articulate the criteria for a function being increasing, decreasing, or constant on a given interval.

2

Apply the definitions of increasing, decreasing, and constant functions to analyze graphs and real-world problems.

3

Identify relative maximum and minimum values from function graphs and understand their significance.

4

Interpret and work with piecewise functions, including determining their domains, ranges, and transition points.

5

Utilize slope information from calculus to distinguish between increasing, decreasing, and constant behavior.

Key Concepts

CONCEPT

DEFINITION

Increasing Function

A function f is increasing on an open interval I if for any two numbers a and b in I with a < b, f(a) ≤ f(b) (and typically f(a) < f(b)).

Decreasing Function

A function f is decreasing on an open interval I if for any two numbers a and b in I with a < b, f(a) ≥ f(b) (and typically f(a) > f(b)).

Constant Function

A function f is constant on an interval I if f(a) = f(b) for all a and b in I.

Relative Maximum

A function f has a relative maximum at a point c if f(c) is greater than all nearby function values in some open interval around c.

Relative Minimum

A function f has a relative minimum at a point c if f(c) is less than all nearby function values in some open interval around c.

Slope

In calculus, the slope of the tangent line to the function at a point indicates whether the function is increasing (positive slope), decreasing (negative slope), or constant (zero slope).

Example Problems

Example 1

Determine the intervals on which the function is (a) increasing (b) decreasing; (c) constant. THE GRAPH CANNOT COPY

Example 2

Determine the intervals on which the function is (a) increasing (b) decreasing; (c) constant. THE GRAPH CANNOT COPY

Example 3

Determine the intervals on which the function is (a) increasing (b) decreasing; (c) constant. THE GRAPH CANNOT COPY

Example 4

Determine the intervals on which the function is (a) increasing (b) decreasing; (c) constant. THE GRAPH CANNOT COPY

Example 5

Determine the intervals on which the function is (a) increasing (b) decreasing; (c) constant. THE GRAPH CANNOT COPY
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Step-by-Step Explanations

QUESTION

Given a function f with domain (-∞, ∞) and a graph that shows f(x) increasing as x moves from -∞ to -0.082, explain how to determine the interval on which f is increasing.

STEP-BY-STEP ANSWER:

Step 1: Identify the portion of the domain under consideration. In this example, observe the graph as x increases from negative infinity to a specific point (e.g., -0.082).
Step 2: Check the outputs f(a) and f(b) for any a and b within that interval with a < b, confirming that f(a) ≤ f(b) and, ideally, f(a) < f(b).
Step 3: Conclude that since the y-values consistently increase, the function is increasing on that identified interval.
Final Answer: The function is increasing on the interval where x is less than or equal to -0.082 (as determined by the specific graph details).

Determining an Increasing Interval

QUESTION

For a function given by f(x) = 0.1x³ - 0.6x² - 0.1x + 2, explain the process to identify its relative maximum and minimum points from its graph.

STEP-BY-STEP ANSWER:

Step 1: Analyze the graph to locate the peak (highest point) and the valley (lowest point) on the curve. These are the potential relative maximum and minimum, respectively.
Step 2: Verify by observing that at the peak, the function transitions from increasing to decreasing, while at the valley, it transitions from decreasing to increasing.
Step 3: Read or calculate the corresponding y-values at these transition points (for example, a y-value around 2.004 could represent a relative maximum and a y-value around -1.604 a relative minimum).
Final Answer: The relative maximum and minimum are determined through graphical analysis by identifying where the function changes its increasing/decreasing behavior.

Identifying Relative Extrema

QUESTION

How do you evaluate a piecewise function f(x) defined by different formulas on different parts of its domain for a given x value?

STEP-BY-STEP ANSWER:

Step 1: Determine which interval or 'piece' of the function the input x belongs to by examining the domain restrictions.
Step 2: Use the formula corresponding to that interval to calculate f(x).
Step 3: If needed, check for continuity or any jumps at the boundary points between pieces.
Final Answer: The value of f(x) is found by matching x to the correct piece of the function and applying the corresponding formula.

Analyzing Piecewise Functions

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Common Mistakes

  • Misinterpreting endpoints when using open interval notation, which may lead to incorrectly labeling the function as both increasing and decreasing at a point.
  • Confusing the definitions of increasing and decreasing, especially by not rigorously checking the condition f(a) < f(b) for all a < b.
  • Overlooking the implications of a slope being zero, which indicates a constant interval rather than a transition point.
  • Ignoring the domain restrictions of piecewise functions and applying the wrong formula for a given input.