Variables and Relationships

Now that we have studied the various types of variables, we can learn to describe the relationships between variables. There are three(3) types of relationships:

1. Direct relationship: If we increase one variable then the other variable increases. If we decrease one, the other decreases.
2. Inverse relationship: Increasing one variable decreases the other
3. No relationship: Increasing/decreasing one variable shows no noticeable change in the other variable

What do they look like?

If we plot graphs of one variable versus the other then we can tell the relationship by looking at the best fit line/regression line/trend line.

• Direct relationships produce a best fit line with a positive gradient (resembles a forward slash, $/$)

• Inverse relationships produce a best fit line with a negative gradient (resembles a backward slash, $\backslash$)

• When there is no relationship, the best fit line can be horizontal (gradient is zero) or vertical (gradient is undefined)

Recap on finding gradient

The gradient is the ratio of the rise to the run. It is the rate of change of $y$ with respect to $x$: $$ \begin{equation}\begin{aligned} gradient(m)=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\\ \end{aligned}\end{equation} $$ e.g. the gradient of the line segment connecting $(1,2)$ and $(3,8)$ is $$ \begin{equation}\begin{aligned} gradient=\frac{y_2-y_1}{x_2-x_1}=\frac{8-2}{3-1}=\frac{6}{2}=3\\ \end{aligned}\end{equation} $$ This means for every unit increase in $x$, we will see a 3 unit increase in $y$.

Mission details

Find the gradient of the graphs with the points:

1. $(1,3)$ and $(2,5)$
2. $(4,2)$ and $(5,1)$
3. $(3,2)$ and $(5,2)$
4. $(2,1)$ and $(2,7)$