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Chapter 2: Problem 89

A home owner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period beginning on a Sunday.

Short Answer

Expert verified
The graph should show a repeating weekly cycle with steady increases and sudden drops every Wednesday.

Step by step solution

01

Understand the Problem

We need to sketch the grass height as a function of time over four weeks, starting on Sunday. The grass grows continuously, but it is cut every Wednesday.
02

Plot Key Points

Identify the key events: the grass starts growing on Sunday, reaches its peak on Wednesday before mowing, and then drops sharply post-mowing. This pattern repeats weekly.
03

Consider Growth and Mowing

Sketch the graph such that it increases steadily from Sunday to Wednesday, symbolizing grass growth. On Wednesday, depict a sudden drop in height due to mowing.
04

Repeat the Weekly Pattern

Extend the pattern for the next three weeks: repeat the growth from Sunday to Wednesday and the mowing on each Wednesday, showing continuous growth and sharp decreases.
05

Finalize the Sketch

Ensure the graph shows four repeating cycles, each beginning with an increase from Sunday to Wednesday and a decrease on Wednesday, completing each weekly cycle by Saturday.

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Key Concepts

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

Understanding Periodic Functions
Periodic functions have values that repeat at regular intervals, much like the pattern of grass growth and cutting in our exercise. When a function is periodic, its graph will show distinct cycles repeated over time. For example, with the homeowner mowing the lawn every Wednesday, each week forms one full cycle.

In this case, from Sunday to Wednesday, the grass height increases steadily. Then, the height falls sharply on Wednesday after mowing. This represents the high-low sequence typical in periodic functions.

To create a graph of a periodic function:
  • Identify the repeating interval (in this case, weekly).
  • Determine the key points within the cycle (start, peak, and end).
  • Sketch the repeating pattern over the defined period (four weeks for our graph).
Understanding periodic functions can help visualize repetitive real-world situations effectively.
Exploring Piecewise Functions
A piecewise function is a function broken up into "pieces," with each piece representing a different behavior based on its particular formula. In the context of our lawn mowing scenario, the grass height can be represented by a piecewise function.

Each week can be divided into two main pieces:
  • One piece where the grass height gradually increases (from Sunday to Wednesday morning). This is often modeled with a linear function due to the steady growth rate of the grass.
  • Another piece that shows a sudden drop on Wednesday afternoon after mowing, depicting an immediate decrease to a lower height.
When sketching a piecewise function on a graph, ensure the transitions (in this case, mowing on Wednesday) are marked clearly, illustrating how the behavior changes at specified times.

Piecewise functions are valuable in representing situations where distinct events change outcomes over specific intervals.
Real-World Applications of Functions
Functions are everywhere in the real world, and they help us describe a variety of phenomena. The grass mowing problem is a practical example that can be extended to numerous other applications:

  • Weather patterns: Temperature might change predictably each day before cooling at night, similar to a periodic function.
  • Economy: Stock prices often follow trends based on similar market behaviors captured by piecewise functions.
  • Industrial processes: Machines may operate continuously at fixed rates but then periodically pause for maintenance, mimicking periodic functions with distinct shifts like piecewise functions.
By applying the concept of functions to real-world scenarios, students gain insights into how mathematical models can predict and analyze repetitive or situational changes.

Understanding real-world applications can enhance problem-solving skills and lead to creative solutions in everyday life.

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Most popular questions from this chapter

In a certain country the tax on incomes less than or equal to \(鈧� 20,000\) is \(10 \% .\) For incomes that are more than \(鈧� 20,000\) the tax is \(鈧� 2000\) plus \(20 \%\) of the amount over \(鈧� 20,000.\) (a) Find a function \(f\) that gives the income tax on an income \(x\). Express \(f\) as a piecewise defined function. (b) Find \(f^{-1}\). What does \(f^{-1}\) represent? (c) How much income would require paying a tax of \(鈧� 10,000 ?\)

Find a function whose graph is the given curve. The bottom half of the circle \(x^{2}+y^{2}=9\)

A linear function is given. (a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$f(x)=\frac{1}{2} x+3$$

The table gives the population in a small coastal community for the period 1997-2006. Figures shown are for January 1 in each year. (a) What was the average rate of change of population between 1998 and 2001 ? (b) What was the average rate of change of population between 2002 and 2004 ? (c) For what period of time was the population increasing? (d) For what period of time was the population decreasing? $$\begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1997 & 624 \\ 1998 & 856 \\ 1999 & 1,336 \\ 2000 & 1,578 \\ 2001 & 1,591 \\ 2002 & 1,483 \\ 2003 & 994 \\ 2004 & 826 \\ 2005 & 801 \\ 2006 & 745 \\ \hline \end{array}$$

Graphing Functions Sketch a graph of the function by first making a table of values. $$C(t)=-\frac{1}{t+1}$$
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