Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $6 \leq x \leq 7$?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 6 \) and \( b = 7 \).

Step2: Find \( f(6) \) and \( f(7) \) from the graph

From the graph, at \( x = 6 \), the point is on the x - axis, so \( f(6)=0 \). At \( x = 7 \), we need to determine the y - value. Looking at the graph, between \( x = 6 \) (where \( y = 0 \)) and \( x = 8 \) (where \( y = 4 \)), we can see the pattern. Let's assume the grid is such that we can find the value. Wait, actually, looking at the graph, at \( x = 6 \), \( f(6)=0 \), and at \( x = 7 \), let's check the coordinates. Wait, maybe I misread. Wait, the graph has a point at \( x = 6 \) (y = 0), at \( x = 7 \), let's see the curve. Wait, maybe the graph has a point at \( x = 7 \) with y - value? Wait, no, maybe I made a mistake. Wait, the average rate of change formula is \(\frac{f(7)-f(6)}{7 - 6}\). From the graph, \( f(6) = 0 \) (since it's on the x - axis at \( x = 6 \)), and at \( x = 7 \), let's see the graph. Wait, maybe the point at \( x = 7 \) is such that we can calculate. Wait, no, maybe the graph: at \( x = 6 \), \( f(6)=0 \), at \( x = 7 \), let's check the y - value. Wait, maybe I made a mistake. Wait, the interval is \( 6\leq x\leq7 \). Let's re - examine the graph. The graph has a point at \( x = 6 \) (y = 0), and then between \( x = 6 \) and \( x = 8 \), the graph goes from (6,0) to (8,4). Wait, but we need \( x = 7 \). Wait, maybe the function is a parabola or something? Wait, no, the average rate of change is \(\frac{f(7)-f(6)}{7 - 6}\). Let's find \( f(6) \) and \( f(7) \). From the graph, \( f(6)=0 \). Now, what is \( f(7) \)? Let's assume that the graph from \( x = 6 \) to \( x = 8 \) is a line? Wait, no, but maybe the coordinates: at \( x = 6 \), \( y = 0 \); at \( x = 7 \), let's see, if we look at the grid, maybe the point at \( x = 7 \) has \( y = 1 \)? No, that doesn't make sense. Wait, maybe I misread the graph. Wait, the graph has a point at \( x = 6 \) (0,0)? No, \( x = 6 \), \( y = 0 \). At \( x = 7 \), let's see, the graph is increasing from \( x = 6 \) (y = 0) to \( x = 8 \) (y = 4). So the slope between \( x = 6 \) and \( x = 8 \) is \(\frac{4 - 0}{8 - 6}=\frac{4}{2}=2\). But we need between \( x = 6 \) and \( x = 7 \). Wait, maybe the function is linear between \( x = 6 \) and \( x = 8 \)? If that's the case, then the slope (average rate of change) between \( x = 6 \) and \( x = 7 \) would be the same as between \( x = 6 \) and \( x = 8 \) if it's linear. Wait, no, if it's linear, the slope is constant. So if from \( x = 6 \) (y = 0) to \( x = 8 \) (y = 4), the slope is \(\frac{4 - 0}{8 - 6}=2\). But we need \( x = 6 \) to \( x = 7 \), so the average rate of change is \(\frac{f(7)-f(6)}{7 - 6}\). If the function is linear between \( x = 6 \) and \( x = 8 \), then \( f(7)=f(6)+(7 - 6)\times2=0 + 2=2 \). So \( f(7)=2 \), \( f(6)=0 \). Then the average rate of change is \(\frac{2 - 0}{7 - 6}=\frac{2}{1}=2\). Wait, but maybe I made a mistake. Wait, let's do it properly. The formula for average rate of change is \(\text{Average Rate of Change}=\frac{f(b)-f(a)}{b - a}\), where \( a = 6 \), \( b = 7 \). So we need \( f(6) \) and \( f(7) \). From the graph, at \( x = 6 \), the point is (6,0), so \( f(6)=0 \). At \( x = 7 \), let's look at the graph. The graph goes from (6,0) to (8,4). So the change in y is \( 4 - 0 = 4 \) over a change in x of \( 8 - 6 = 2 \), so the slope (rate of change) per unit x is \( \frac{4}{2}=2 \). So between \( x = 6 \) and \( x = 7 \) (a change in x of 1), the change in y is 2. So \( f(7)=f(6)+2=0 + 2 = 2 \). Then the average rate of change is \(\frac{f(7)-f(6)}{7 - 6}=\frac{2 - 0}{1}=2\).

Answer:

2