Question

Two students in your class, Tucker and Karly, are disputing a function. Tucker says that for the function, between x = -3 and x = 3, the average rate of change is 0. Karly says that for the function, between x = -3 and x = 3, the graph goes up through a turning point, and then back down. Explain how Tucker and Karly can both be correct, using complete sentences.

Answer 1

Any ideas?

Answer 2

I believe that another name for the 'rate of change' of any function is also the 'slope'. Given any two points, (x1, y1) and (x2, y2), the slope or rate of change of the function between them is given as:
\[Slope = \frac{ y _{2}- y _{1} }{ x _{2} -x_{1} }\]
From this formula, one can conclude that the rate of change (i.e. the slope of the line between the start and end points) will be 0 if the numerator is 0. For this to happen the y-coordinates of the two points must be the same (y1 = y2)

Let's consider our start and end points to be the points at which the function cuts the lines x= -3 and x = 3. To answer this question, it doesn't really matter what we choose our y1 = y2 value to be, but for simplicity I will choose it to be at 0. So, our start and end points will be at (-3, 0) and (3, 0)


If we were to apply this to the start and end points of (-3, 0) and (3, 0), we would end up with an answer of 0, indicating that if we were to draw an imaginary line between them, it would be flat/parallel with the x-axis. We would never usually be interested in the slope of a curved function such as this based on the start and end points, as they give us no idea as to the trend/shape of the function in general. A slope is more useful in describing the trend of linear functions in this regard, as it does not take into account this turning point or indeed any feature in between the start and end points.

So, based on this formula we can see how the rate of change between the 'start' and 'end' points we are considering may be 0, no matter what it does between these values of x = -3 and x = 3.