3.08 Descriptions, tables, equations, and graphs
Table of values, equation and graph
Linear relationships can be represented in a table, in an equation, verbally, or graphically.
Let's say the temperature of a solution increases by 3\degree C every minute. If the initial temperature of the solution is -5\degreeC, we can model the relationship between the temperature of the solution and the time it takes to heat the solution.
We can represent the temperature as y and the number of minutes as x so that we have an equation y=3x-5.
An equation in the form of y=mx+b or y=mx represents a linear function. From an equation, we can create a table of values represented by two quantities (usually x and y) that are related in some way.
If we want to construct a table of values for the equation:
y=3x-5
we can substitute the values of x into the equation and complete the table by solving to find the y values. Let's look at the following table:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y |
Substitute x=1:
| \displaystyle y | \displaystyle = | \displaystyle 3(1)-5 | Substitute the value of x |
| \displaystyle = | \displaystyle 3-5 | Evaluate the multiplication | |
| \displaystyle = | \displaystyle -2 | Evaluate |
Substitute x=2:
| \displaystyle y | \displaystyle = | \displaystyle 3(2)-5 | Substitute the value of x |
| \displaystyle = | \displaystyle 6-5 | Evaluate the multiplication | |
| \displaystyle = | \displaystyle 1 | Evaluate |
Substitute x=3:
| \displaystyle y | \displaystyle = | \displaystyle 3(3)-5 | Substitute the value of x |
| \displaystyle = | \displaystyle 9-5 | Evaluate the multiplication | |
| \displaystyle = | \displaystyle 4 | Evaluate |
Substitute x=4:
| \displaystyle y | \displaystyle = | \displaystyle 3(4)-5 | Substitute the value of x |
| \displaystyle = | \displaystyle 12-5 | Evaluate the multiplication | |
| \displaystyle = | \displaystyle 7 | Evaluate |
The completed table of values is:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | -2 | 1 | 4 | 7 |
Each column in a table of values may be grouped together in the form (x,y) , to create an ordered pair. We can plot each ordered pair as a point on the coordinate plane, and then draw the line through the points.
The table of values has the following ordered pairs:\left(1,-2\right),\left(2,1\right),\left(3,4\right),\left(4,7\right)
We can plot each ordered pair as a point on the coordinate plane.
To complete the graph of the equation y=3x-5 we will connect the points that we graphed with a straight line.
Examples
Example 1
Consider the equation y=3x+1.
Complete the table of values below:
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
Plot the points in the table of values
Draw the graph of y=3x+1.
A linear function can be represented by an equation, a table of values and graph.
To complete a table of values substitute each value of x into the equation to find the value of y.
Each column in a table of values may be grouped together as an ordered pair (x,y).We can plot each ordered pair as a point on the coordinate plane, and then draw the line through the points graphed.