4. Proportional Relationships & Lines

United States of America

Grade 8

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.

Consider the equation y=3x+1.

a

Complete the table of values below:

x | -1 | 0 | 1 | 2 |
---|---|---|---|---|

y |

Worked Solution

b

Plot the points in the table of values

Worked Solution

c

Draw the graph of y=3x+1.

Worked Solution

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.