Question

to the right, you will see the graph of a function. ignoring whether it is increasing or decreasing, which interval has the greatest average rate of change?
○ x = 0 and x = 5
○ x = 1 and x = 5
○ x = 2 and x = 4
○ x = 2 and x = 6
done

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$, which is the slope of the secant line connecting the points $(a,f(a))$ and $(b,f(b))$. Geometrically, we can estimate the vertical change (rise) and horizontal change (run) for each interval.

Step2: Analyze interval $x = 0$ and $x = 5$

By looking at the graph, estimate the $y$-values at $x = 0$ and $x = 5$. Let's assume $y_1$ at $x = 0$ and $y_2$ at $x = 5$. The run is $5-0=5$.

Step3: Analyze interval $x = 1$ and $x = 5$

The run is $5 - 1=4$. Estimate the $y$-values at $x = 1$ and $x = 5$ and calculate the rise.

Step4: Analyze interval $x = 2$ and $x = 4$

The run is $4 - 2=2$. Estimate the $y$-values at $x = 2$ and $x = 4$. Since the graph is steeper in this short - interval compared to the others in terms of the ratio of rise to run.

Step5: Analyze interval $x = 2$ and $x = 6$

The run is $6 - 2=4$. Estimate the $y$-values at $x = 2$ and $x = 6$.
The secant line for the interval $x = 2$ and $x = 4$ has the steepest slope among the given intervals.

Answer:

$x = 2$ and $x = 4$