Question

1. $1 < x < 2$ determine average rates of change | wayground find the average rate of change over the given interval. a) -2 b) -0.5 c) -1 d) 2

Explanation:

Step1: Identify points on the interval

From the graph, when \( x = 1 \), \( y = 3 \) (point \((1, 3)\)); when \( x = 2 \), \( y = 1 \) (point \((2, 1)\)).

Step2: Apply average rate of change formula

The formula for average rate of change over \([a, b]\) is \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 1 \), \( b = 2 \), \( f(1) = 3 \), \( f(2) = 1 \).
So, \(\frac{1 - 3}{2 - 1} = \frac{-2}{1} = -2\). Wait, no, wait—wait, maybe I misread the graph. Wait, let's check again. Wait, maybe the points are \((1, 3)\) and \((3, 1)\)? No, the interval is \(1 < x < 2\), so \(x = 1\) and \(x = 2\). Wait, maybe the y - value at \(x = 1\) is 3, at \(x = 2\) is 1? Wait, no, maybe I made a mistake. Wait, let's recalculate. Wait, average rate of change is \(\frac{\Delta y}{\Delta x}\). So \(\Delta y = y_2 - y_1\), \(\Delta x = x_2 - x_1\). If \(x_1 = 1\), \(y_1 = 3\); \(x_2 = 2\), \(y_2 = 1\), then \(\frac{1 - 3}{2 - 1}=\frac{-2}{1}=-2\)? But that's option a. Wait, but maybe the points are different. Wait, maybe at \(x = 1\), \(y = 3\) and at \(x = 3\), \(y = 1\)? No, the interval is \(1 < x < 2\), so \(x\) from 1 to 2. Wait, maybe the graph has a point at (1, 3) and (2, 1)? Then the average rate of change is \(\frac{1 - 3}{2 - 1}=-2\), which is option a. But wait, maybe I misread the y - axis. Wait, let's check the grid. Each grid square—maybe the y - value at \(x = 1\) is 3, at \(x = 2\) is 1. So the calculation is \(\frac{1 - 3}{2 - 1}=-2\).

Wait, no, wait, maybe the points are (1, 3) and (2, 1), so the average rate of change is \(\frac{1 - 3}{2 - 1}=-2\), which is option a.

Answer:

a) -2