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Chapter 1: Problem 46

Sketch a reasonable graph for each of the following functions. [UW] a. Distance of your big toe from the ground as you ride your bike for 10 seconds. b. Your height above the water level in a swimming pool after you dive off the high board. c. The percentage of dates and names you'll remember for a history test, depending on the time you study.

Short Answer

Expert verified
Function A is a sinusoidal wave, Function B is an inverted parabola, and Function C is a logarithmic curve.

Step by step solution

01

Analyze Function A

For part (a), consider the distance of your big toe from the ground as you ride your bike. As you pedal, the toe's motion is periodic due to the circular motion of the pedals. This results in a sinusoidal graph representing the distance over time. The peaks and troughs correspond to the pedal's highest and lowest points.
02

Sketch Function A

Draw the x-axis representing time in seconds (0 to 10 seconds) and the y-axis representing distance from the ground. Sketch a sinusoidal wave starting at the average distance, peaking near the top (maximum distance) and troughing at the bottom (minimum distance), repeating this for a few cycles.
03

Analyze Function B

For part (b), consider your height above the water when diving. Initially, your height increases as you jump upwards, then decreases rapidly as you fall towards the water. After entering the water, your height remains constant at zero (the water level). This sketch will resemble an inverted parabola with a sharp descent.
04

Sketch Function B

Draw the x-axis for time and the y-axis for height above the water. Start with an increase in height as you jump upwards, reaching a peak, followed by a steep decrease as you dive. Once you hit the water, continue a straight line along the x-axis representing zero height.
05

Analyze Function C

For part (c), consider the percentage of dates/names you remember for a history test based on studying time. Initially, memory improves slowly with study time, but as you study more and approach your capability, the rate of improvement slows down, leading to a saturation point, indicating a logarithmic graph.
06

Sketch Function C

Draw the x-axis as study time and the y-axis as percentage remembered. Start with a gentle, upward slope that increases rapidly initially but slows as it approaches a maximum percentage, forming a curve that flattens, representing diminishing returns with more study time.

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Key Concepts

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

Sinusoidal Graph
When considering the motion of your big toe as you pedal a bicycle, we can model this visually using a sinusoidal graph. This type of graph is ideal for representing periodic functions, which repeat at regular intervals. Imagine the pedals versus time:
  • The toe moves up and down constantly as the pedals turn.
  • This results in a cyclic pattern where the toe's distance from the ground oscillates.
The sinusoidal shape, similar to a sine wave, captures this rhythmic up and down pattern:
  • The peak of the sine wave represents the point where the toe is at its furthest distance from the ground.
  • Conversely, the trough represents when the toe is closest to the ground.
  • The midline of this wave corresponds to the average distance from the ground.
Understanding this helps in predicting motion behaviors which are cyclical, an essential skill in physics and engineering.
Inverted Parabola
Visualizing a dive from a high board can be best described through an inverted parabola graph. An inverted parabola captures the journey of height above water level:
  • Initially, you push off from the board, ascending slightly as you gain height.
  • Gravity then kicks in, pulling you rapidly down towards the water.
  • Upon hitting the water, your height stabilizes at zero above the water level.
This graph is often shaped like an upside-down 'U', representing the rapid decrease after reaching the maximum height:
  • The peak of the parabola is your highest point, right before the descent.
  • The rapid descent represents the swift dive into the pool.
  • After entering the water, your graph continues flat along the x-axis at zero, indicating the constant height underwater.
Thus, inverted parabolas are useful for modeling scenarios with rapid ascents and descents, such as freefalls.
Logarithmic Graph
A logarithmic graph effectively represents the relationship between study time and the percentage of dates or names recalled. This graph depicts the gain in memory over time:
  • Initially, you recall information at a faster rate as you start studying.
  • Over time, however, the rate of memory gains slows down as you move closer to your cognitive limits.
  • This reflects the concept of diminishing returns, where extra study time yields smaller increases in memory.
The curve of a logarithmic graph starts steep and flattens over time:
  • Initially, there is a steep upward curve because learning is swift when new to the study material.
  • As you become familiar, the curve levels off, reflecting reduced memory gain.
  • The tail end of the graph shows a near plateau, highlighting minimal gains despite additional study efforts.
Logarithmic graphs are particularly helpful in scenarios where growth scales decline after a certain point, prevalent in learning and natural growth processes.

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Most popular questions from this chapter

Sketch a graph of each function as a transformation of a toolkit function. $$ h(x)=|x-1|+4 $$

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