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4.07 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:

x1234
y

Substitute x=1:

\displaystyle y\displaystyle =\displaystyle 3(1)-5Substitute the value of x
\displaystyle =\displaystyle 3-5Evaluate the multiplication
\displaystyle =\displaystyle -2Evaluate

Substitute x=2:

\displaystyle y\displaystyle =\displaystyle 3(2)-5Substitute the value of x
\displaystyle =\displaystyle 6-5Evaluate the multiplication
\displaystyle =\displaystyle 1Evaluate

Substitute x=3:

\displaystyle y\displaystyle =\displaystyle 3(3)-5Substitute the value of x
\displaystyle =\displaystyle 9-5Evaluate the multiplication
\displaystyle =\displaystyle 4Evaluate

Substitute x=4:

\displaystyle y\displaystyle =\displaystyle 3(4)-5Substitute the value of x
\displaystyle =\displaystyle 12-5Evaluate the multiplication
\displaystyle =\displaystyle 7Evaluate

The completed table of values is:

x1234
y-2147

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.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
-1
1
2
3
4
5
6
7
8
y

To complete the graph of the equation y=3x-5 we will connect the points that we graphed with a straight line.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
-1
1
2
3
4
5
6
7
8
y

This straight line is the graph of y=3x-5 which we used to complete the table of values.

We can see that the y values continuously increase as the x values increase.

We can say that the function is increasing.

Examples

Example 1

Consider the equation y=3x+1.

a

Complete the table of values below:

x-1012
y
Worked Solution
Create a strategy

Substitute each values from the tables into the given equation.

Apply the idea

For x=1:

\displaystyle y\displaystyle =\displaystyle 3\times (-1)+1Substitute -1 to x
\displaystyle =\displaystyle -2Evaluate

Similarly, if we substitute the other values of x, ( x=0,\, x=1,\, x=2 ), into y=3x+1, we get:

x-1012
y-2 14 7
b

Plot the points in the table of values

Worked Solution
Create a strategy

For an ordered pair (a,b) from the given table of values found in part (a), identify where x=a along the x-axis and y=b along the y-axis.

Apply the idea

Since we are given table of values, then the ordered pairs of points to be plotted on the coordinate plane are (-1,-2),(0,1),(1,4) and (2,7).

-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
6
7
y
c

Draw the graph of y=3x+1.

Worked Solution
Create a strategy

Use the plotted points on the coordinate plane from part (b).

Apply the idea

The equation y=3x+1 must pass through each of the plotted points.

-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
6
7
y
Idea summary

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.

Outcomes

8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.B.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g. Where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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