Video: Comparing the Rate of Change of Two Linear Functions

Which of the following has a greater rate of change? [A] Graph a [B] 𝑦 = 2π‘₯ βˆ’ 9

02:25

Video Transcript

Which of the following has a greater rate of change? Is it graph (a) or (b) 𝑦 equals two π‘₯ 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 π‘šπ‘₯ plus 𝑏. The value of π‘š 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 three.

We can calculate the slope or gradient of the straight line using the formula 𝑦 one minus 𝑦 two divided by π‘₯ one minus π‘₯ two. This is also known as the change in 𝑦 over the change in π‘₯ 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 𝐡 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 π‘₯-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 π‘₯ 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.

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