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Wize High School Grade 12 Pre-Calculus Textbook > Rates of Change

Graphs & Rates of Change

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Graphs & Rates of Change

Remember:
  • Motion: a change in position measured by distance and time.
  • Speed: the rate at which an object moves.
  • Velocity: speed and direction of a moving object
  • Acceleration: the rate speed or direction changes
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Distance - Time Graphs

Plotting distance (y-axis) and time (x-axis) tells us a lot about motion.

  1. When an object is at rest:

  1. When an object is moves at a constant speed:


  1. When an object returns to the start:



  1. When an object accelerates:


In summary, a distance-time graph tells us how far an object has moved with time.
  • The steeper the graph, the faster the motion.
  • A horizontal line means the object is not changing its position. It is not moving. It is at rest.
  • A downward slope means the object is returning to start.
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Speed - Time Graphs

Plotting speed (y-axis) and time (x-axis) graphs are also called velocity - time Graphs.

  1. The moving object is travelling at a constant speed:



  1. The moving object is accelerating:

  1. The moving object is decelerating:

In summary, a speed-time graph shows us how the speed of a moving object changes with time.
  • The steeper the graph, the greater the acceleration.
  • A horizontal line means the object is moving at a constant speed.
  • A downward slope means the object is slowing down.

Example: Graphs & Rates of Change

The following graph is a displacement (m) vs. time (mins) graph of a cyclist.

  1. What is the average rate of change in the first 1010 minutes?
  2. What is the average rate of change in the first 2020 minutes?
  3. What is the instantaneous rate of change at t=3t=3 minutes?
  4. What is the instantaneous rate of change at t=16t=16 minutes?
a.
The coordinates at 10 minutes: (10,2.5)(10,2.5)
The coordinates at 0 minutes: (0,0)(0,0)

mavg=2.5−010−0=0.25 meters/min\begin{array}{rcl} m_{avg}&=&\dfrac{2.5-0}{10-0}\\\\ &=&0.25~\text{meters}/\text{min} \end{array}

b.
The coordinates at 20 minutes: (20,0)(20,0)
The coordinates at 0 minutes: (0,0)(0,0)
mavg=0−020−0=0 meters/min\begin{array}{rcl} m_{avg}&=&\dfrac{0-0}{20-0}\\\\ &=&0~\text{meters}/\text{min} \end{array}

c. The instantaneous rate of change at t=3t=3 minutes is the slope of the line.

The equation of the line can be found using the two coordinate points (4,2)(4,2) and (0,0)(0,0).

m=0−20−4=0.5m=\dfrac{0-2}{0-4}=0.5

The instantaneous rate of change at t=3t=3 minutes is 0.5 meters/min0.5~\text{meters}/\text{min}.


d. The instantaneous rate of change at t=16t=16 minutes is the slope of the line.

The equation of the line can be found using the two coordinate points (12,3)(12,3) and (20,0)(20,0).

m=0−320−12=−0.375m=\dfrac{0-3}{20-12}=-0.375

The instantaneous rate of change at t=16t=16 minutes is −0.375 meters/min-0.375~\text{meters}/\text{min}.


Practice: Graphs & Rates of Change

The distance-time graphs below represent the motion of a semi-truck. Match the descriptions with the graphs.
A.


B.


C.


D.


The car is travelling at a constant speed.
The car is coming back.
The car is stopped.
The speed of the car is decreasing.

Practice: Graphs & Rates of Change

The graph below shows the change in speed of a Greyhound bus (m/s)(\text{m/s}) during a part of its journey.




Practice: Graphs & Rates of Change

A graph displays changes in distance versus time. The graph has a downward sloping line from point to point. If the graph has been drawn to display speed versus time, how would it be different?

Extra Practice
Graphs and Rates of Change
The following is a speed vs. time graph of a car in km/hr\text{km}/\text{hr}
Rates of Change
The following is a distance vs time graph of a cat.
Determine the following:
Graphs & Rates of Change
Practice: Graphs & Rates of Change
A graph displays changes in distance versus time. The graph has a straight horizontal line from point to point. If the graph has been drawn to display speed versus time, how would it be different?
Graphs & Rates of Change
Practice: Graphs & Rates of Change
A graph displays changes in speed versus time. The graph has a downward sloping line from point to point. If the graph has been drawn to display distance versus time, how would it be different?
Graphs & Rates of Change
Practice: Graphs & Rates of Change
The graph below shows the change in speed of a sprinter (m/s)(\text{m/s}) during a part of their journey.
Graphs & Rates of Change
Practice: Graphs & Rates of Change
The graph below shows the change in speed of a motorcylce (m/s)(\text{m/s}) during a part of its journey.
Graphs & Rates of Change
Practice: Graphs & Rates of Change
The distance-time graphs below represent the motion of a runner. Match the descriptions with the graphs.