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Chapter 3: Problem 3

Describe what the rate of change looks like in each graph. a. the graph of a person walking at a steady rate toward a motion sensor (a) b. the graph of a person standing still c. the graph of a person walking at a steady rate away from a motion sensor d. the graph of one person walking at a steady rate faster than another person

Short Answer

Expert verified
Graphs a and c show steady moves, graph b is flat (standing), and graph d has steeper lines for faster motion.

Step by step solution

01

Understanding Rate of Change

The rate of change in a graph refers to how the value of the dependent variable (often distance or position) changes in relation to the independent variable (often time). If we think of a graph as a motion diagram, the rate of change often represents speed or velocity.
02

Identifying Steady Rates

A steady rate of change in a graph means that the graph is a straight line. If a person walks toward a sensor at a steady rate or away from it, the graph will show a constant slope. The line can be upward or downward, depending on the direction of movement relative to the sensor.
03

Analyzing Graph a

For graph (a), where a person walks at a steady rate toward a motion sensor, the rate of change is constant, and the slope is negative. This reflects decreasing distance as the person approaches the sensor.
04

Analyzing Graph b

For graph (b), where a person is standing still, the graph is a horizontal line. This means the rate of change is zero, as there is no movement, and the distance remains constant over time.
05

Analyzing Graph c

For graph (c), a person walking at a steady rate away from the sensor, the rate of change is constant with a positive slope. This indicates increasing distance from the sensor.
06

Analyzing Graph d

For graph (d), one person walking faster than another, both graphs will be straight lines. The graph of the faster-moving person shows a steeper slope, reflecting a greater rate of change in distance over time compared to the slower person.

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Key Concepts

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

Graph Interpretation
Graph interpretation involves understanding how different elements on the graph relate to the movement or activity being described. In the context of rate of change, it is crucial to focus on the slope of the line, which tells us about speed or velocity.
  • A steeper slope indicates a faster movement.
  • A horizontal line suggests no movement.
  • An upward or downward slope indicates movement away from or towards the starting point.
When analyzing a graph, always ask yourself which variable is on each axis. Usually, time is on the horizontal axis, while distance or position is on the vertical axis. This will help you understand whether the distance is increasing, decreasing, or remaining constant over time. Understanding these basics makes it much easier to interpret specific graphs.
Constant Slope
A constant slope on a graph is key in understanding steady movement. When we say a slope is constant, we're referring to a straight line on the graph. This indicates that the rate of change in distance over time does not fluctuate. The slope itself can be calculated by taking the difference in the y-values (distance) and dividing it by the difference in the x-values (time).
  • For a person moving at a constant speed, the graph will show a straight line.
  • If the slope is positive, this means movement away from the starting point.
  • A negative slope signifies movement towards the starting point.
  • A horizontal line, or zero slope, means no movement.
The uniformity of a constant slope provides a clear visual of consistent speed, helping you predict future positions based on the existing graph.
Distance-Time Graph
Distance-time graphs are a common way to illustrate movement over time. These graphs help us visualize how an object's position changes as time progresses. On these graphs:
  • The horizontal axis typically represents time.
  • The vertical axis represents distance.
  • A straight line signifies a constant speed.
By examining the slope of a distance-time graph, we can determine how fast or slow someone is moving. The steeper the line, the faster the speed. A flat line means no movement, indicating that the object or person is stationary. Understanding how to read these graphs is crucial for interpreting real-world scenarios, such as a car's trip or a runner's pace.
Velocity
Velocity is a fundamental concept in the context of rate of change. It represents how fast and in what direction an object is moving. On a graph, velocity is represented by the slope.
  • If the slope is positive, velocity is directed away from the origin point.
  • A negative slope indicates velocity towards the origin point.
  • If the graph line is horizontal, the velocity is zero, meaning no change in position over time.
Unlike speed, which only tells how fast an object is moving, velocity tells both how fast and in what direction. This makes it an essential concept in understanding various movement graphs, helping students see not just how quickly movement occurs, but also in which direction it's headed.

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Most popular questions from this chapter

Consider the expression \(\frac{4(y-8)}{y}\). a. Find the value of the expression if \(y=5\). Make a table to show the order of operations. (a) b. Solve the equation \(\frac{4(y-8)}{3}=8\) by undoing the sequence of operations. (a)

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A multiplicative inverse is a number or expression that you can multiply by something to get a value of 1 . The multiplicative inverse of 4 is \(\frac{1}{4}\) because \(4 \cdot \frac{1}{4}=1\). Give the multiplicative inverse of each number. a. 12 b. \(\frac{1}{6}\) c. \(0.02\) d. \(-\frac{1}{2}\)

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APPLICATION Holly has joined a video rental club. After paying \(\$ 6\) a year to join, she then has to pay only \(\$ 1.25\) for each new release she rents. a. Write an equation in intercept form to represent Holly's cost for movie rentals. (a) b. Graph this situation for up to 60 movie rentals. c. Video Unlimited charges \(\$ 60\) for a year of unlimited movie rentals. How many movies would Holly have to rent for this to be a better deal?
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