A **linear relationship** is used to describe a straight-line relationship between two variables.
It can be expressed as a table of values or a graph where the data points are connected via a straight line.
That includes vertical lines and horizontal lines.

Mathematically, a linear relationship is one that satisfies the equation:

\( y = m(x) + b \)

where \(m\) is the slope and \(b\) is the y-intercept. Recall the slope formula is the change in y over change in x.

Every linear equation is a relationship of x and y values. To create a table of values, we just have to pick a set of x values, substitute them into the equation and evaluate to get the y values. You could also read the points from a graph.

Enter the slope and y-intercept below to create a table of values.

The First differences are the differences between consecutive y-values in tables of values with evenly spaced x-values. When the first difference is constant (the same), the relation is linear.

For exmaple, Jameen has recorded his sleep for the week in the table below.

X(day) | Mon | Tues | Wed | Thur | Fri | Sat | Sun |
---|---|---|---|---|---|---|---|

Y(hours) | 8 | 10 | 11 | 9 | 12 | 9 | 10 |

To calculate the first difference, we subtract the second y-value form the first:

\( 10-8 = 2\)

The difference in the y values are 2,1,-2,3,-3,1. Thus we can determine that the relationship of Jameen's sleep is **non-linear**.

Use first differences to determine if the table below has a linear or non linear relationship

X | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

Y | 10 | 13 | 16 | 19 | 22 | 25 |

Show Answer

A scatter plot shows the relationship between two variables on a graph. The values of one variable appear on the horizontal axis (**independent variable**, typically x),
and the values of the other variable appear on the vertical axis (**dependent variable**, typically y).

For exmaple, Jameen has recorded his sleep for the week. The data can also be presented on a graph.

X(day) | Mon | Tues | Wed | Thur | Fri | Sat | Sun |
---|---|---|---|---|---|---|---|

Y(hours) | 8 | 10 | 11 | 9 | 12 | 9 | 10 |

A scatter plot can be used to identify several different types of relationships between two variables.

- The relationship is
**linear**when the points on a scatter plot follow a somewhat straight line pattern. - The relationship is considered
**non-linear**if the points on a scatterplot follow a pattern but not a straight line - The relationship has
**no correlation**if the points on the scatterplot do not show any pattern.

Linear

Non-Linear

No-Correlation

An **outlier** is a data point that does not follow the general trend.

Two variables vary directly **(Direct Variation)** if they are directly proportional to each other. The relation would have the form \( y = m(x)\) (i.e., \(b=0\)) and the graph passes through the origin (0,0)..

A **partial variation** is one where the the value of the dependent variable depends on both the value of the independent variable and some initial value. they are the form \(y = mx + b \) and the graph never passes through the origin (0,0).

In turn, a variation is **inverse ** if y is expressed as the product of some constant number k and the reciprocal of x, given that k \(\ne\) 0

Direct Variation | Indirect Variation | Inverse Variation |
---|---|---|

\(y=2.5(x)\) | \(y=2.5(x)+4\) | \(y=\cfrac{1}{x}\) |

\(y=\cfrac{1}{2}(x)\) | \(y=\cfrac{3}{2}(x)-1/5\) | \(y=\cfrac{-3}{x}\) |

\(y=x\) | \(y=x-1\) | \(y=\cfrac{5}{x}\) |