Which of the following has a
greater rate of change? Is it graph (a) or (b) 𝑦 equals
two 𝑥 minus nine?
As our graph is a straight line,
this means it is a linear function. It can therefore be written in the
form 𝑦 equals 𝑚𝑥 plus 𝑏. The value of 𝑚 is the slope or
gradient of the line, and 𝑏 is the 𝑦-intercept. As the graph crosses the 𝑦-axis at
the point with coordinates zero, three, it is clear that the 𝑦-intercept is
We can calculate the slope or
gradient of the straight line using the formula 𝑦 one minus 𝑦 two divided by 𝑥
one minus 𝑥 two. This is also known as the change in
𝑦 over the change in 𝑥 or the rise over the run. In order to calculate the slope, we
need to select two points that lie on the line. In this case, we’ll choose point 𝐴
with coordinates one, five and point 𝐵 zero, three. It would be just as easy to select
any of the other three points that are labeled: negative one, one; negative two,
negative one; or negative three, negative three.
Subtracting the 𝑦-coordinates
gives us five minus three. Subtracting the 𝑥-coordinates in
the same order gives us one minus zero. This is equal to two divided by
one, which is equal to two. The equation of the straight line
graph is 𝑦 equals two 𝑥 plus three. We were asked to decide which of
the two functions had the greater rate of change. The rate of change is the same as
the slope. As both of the equations have a
slope or gradient of two, they will also have a rate of change of two. We can therefore conclude that the
two functions have the same rate of change.
We can go one step further here and
say that any two functions with the same rate of change or slope will be parallel
lines. This means that they will never
cross or intersect.