what must be true about the average rate of change between any two points on the graph of an increasing…

Answer:

The average rate of change between any two points on the graph of an increasing function is positive.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function $y = f(x)$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Define an increasing function

For an increasing function, if $x_2>x_1$, then $f(x_2)>f(x_1)$. So $f(x_2)-f(x_1)>0$ and $x_2 - x_1>0$.

Step3: Determine the sign of the average rate of change

Since both the numerator $f(x_2)-f(x_1)$ and the denominator $x_2 - x_1$ are positive when considering two points on an increasing - function with $x_2>x_1$, the quotient $\frac{f(x_2)-f(x_1)}{x_2 - x_1}>0$.