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We use graphs to represent data in an easily comprehensible manner. We will learn the basic concepts of graphs, different types of graphs, and its applications with the help of some solved examples. ...Read MoreRead Less
A graph is a representation of the relationship between two or more sets of numbers or measurements.
A graph is also a series of distinct dots, lines, bars, or other symbols that represent a system of connections or inter-relationships between two or more quantities or datasets.
Coordinates are the pairs of numbers that specify the position or location of a point or an object in a two-dimensional coordinate plane or what is known as the “Cartesian” plane.
The X and Y coordinates are a type of address that can be used to locate a point in two dimensions. A point (x,y) refers to any point in the coordinate plane, where the “x” value represents the point’s position relative to the x-axis and the “y” value represents the point’s position relative to the y-axis. The x and y coordinates of a point can be positive or negative depending on where the point is located in the quadrants.
Many changes in the value of distant quantities are dependent on changes in the values of other quantities in today’s world. For example, as the number of families visiting a restaurant grows, so does the restaurant’s revenue; conversely, as the US population grows or shrinks, so will the time it takes to complete a task decrease or increase.
As a result, in some cases, the value of one quantity increases when the value of another quantity decreases, while in other cases, the value of one quantity decreases when the value of another quantity increases.
A graph relating two quantities is increasing if the increase in one quantity is accompanied by an increase in the other.
Linearly Increasing Graph | Increasing at increasing rate graph |
Graph increases linearly | Graph does not increases linearly |
Slope of the graph is the constant | Slope of the graph will increase at the given rate |
Steepness of the graph remains constant | Steepness of the graph will increase at the given rate |
If one quantity increases while the other decreases, the graph relating to the two quantities is decreasing.
Linearly Decreasing graph | Decreasing at an increasing rate graph |
Graph decreases linearly | Graph is not decreasing linearly |
Slope of the graph is a constant | Slope of the graph is decreasing at an increasing rate |
Steepness of the graph remains constant | Steepness of the graph decreases at an increasing rate |
A graph relating two quantities has a constant relationship if the other quantity does not change with the increase in one quantity, i.e. remains constant.
Example 1:
The graphs show the water level in America throughout the day.
Describe the change in water level in America.
Solution:
The water level in America increases at the beginning of the day. The rate of increase slows until the water level begins to decrease. Then the water level decreases at a faster and faster rate for the rest of the day.
Example 2:
The graphs show the water level in Cameroon throughout the day.
Describe the change in water level in Cameroon.
Solution:
The water level decreases at a constant rate at the beginning of the day. Then the water level stays the same for a while before increasing at a constant rate for the rest of the day.
Example 3:
The graph shows the volume of the balloon throughout the day.
Describe the change in volume of the balloon.
Solution:
The volume of the balloon increases at a constant rate at the beginning of the day. Then the volume stays the same for a while before decreasing at a constant rate for the rest of the day.
Example 4:
The graph shows the distances traveled by two bikers in their journey from start to their destination. Describe the speed of each biker throughout their journey. Then determine who finishes first.
Solution:
Biker A | Biker B |
---|---|
The red line increases at a constant rate at the beginning of the journey, stays horizontal for a short time in the middle, and then increases at a constant rate at the end of the journey. | The blue line increases at a constant rate for most of the race, and then increases at a faster and faster rate at the end of the race. |
So, biker A starts running at a constant speed, stops to rest, and then continues to move at a constant rate. | So, Biker B starts moving at a constant speed and continues to move that speed for most of the race. Near the end of the race, Biker B accelerates through the finish line. |
The graph shows that Biker B travels the same distance as Biker A, but in a shorter amount of time. So, Biker B wins the race.
Example 5:
As table size increases, the price increases at an increasing rate.
Solution:
Step 1: Draw the axes. Label the vertical axis “Price” and the horizontal axis “Table size.”
Step 2: Sketch the graph.
The price increases at an increasing rate. So, the graph is nonlinear and becomes steeper as the table size increases.
From a given point in the coordinate axes, the x and y coordinates can be easily determined. The first value is always the x coordinate, and the second value is always the y coordinate for a point (a, b).
A graph relating two quantities is increasing if the increase in one quantity is accompanied by an increase in the other.
If one quantity increases while the other decreases, a graph relating the two quantities is decreasing.