Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-3 \leq x \leq 8$?
answer attempt 1 out of 2

Explanation:

Step1: Find f(-3) and f(8)

From the graph, when \( x = -3 \), \( f(-3) = -20 \) (by looking at the grid, the point at \( x = -3 \) has a y - value of -20). When \( x = 8 \), \( f(8)=30 \) (the point at \( x = 8 \) has a y - value of 30).

Step2: Use the average rate of change formula

The formula for the average rate of change of a function \( y = f(x) \) on the interval \([a,b]\) is \( \frac{f(b)-f(a)}{b - a} \). Here, \( a=-3 \), \( b = 8 \), \( f(a)=f(-3)=-20 \), \( f(b)=f(8)=30 \).

Substitute the values into the formula: \( \frac{f(8)-f(-3)}{8-(-3)}=\frac{30-(-20)}{8 + 3}=\frac{30 + 20}{11}=\frac{50}{11}\approx4.545 \) (but we'll keep it as a fraction for accuracy). Wait, wait, maybe I misread the graph. Let me re - check. Wait, maybe at \( x=-3 \), the y - value: looking at the graph, when \( x=-3 \), the point is at \( y=-20 \)? Wait, no, maybe I made a mistake. Wait, let's re - examine the graph. Wait, the left - hand side: when \( x=-3 \), the graph is at \( y = - 20 \)? Wait, no, maybe the correct values: Let's see, the vertex on the left: when \( x=-3 \), maybe \( f(-3)=-20 \), and at \( x = 8 \), \( f(8)=30 \). Then the average rate of change is \( \frac{30-(-20)}{8-(-3)}=\frac{50}{11}\approx4.55 \). Wait, but maybe I got the y - values wrong. Wait, let's check again. Wait, the function at \( x=-3 \): looking at the grid, each square is, say, 10 units? Wait, the y - axis has marks at 0, 20, 40, etc. Wait, when \( x=-3 \), the point is at \( y=-20 \), and at \( x = 8 \), the point is at \( y = 30 \). So the difference in y is \( 30-(-20)=50 \), the difference in x is \( 8-(-3)=11 \), so the average rate of change is \( \frac{50}{11}\approx4.55 \). But wait, maybe I misread the graph. Wait, another way: maybe at \( x=-3 \), \( f(-3)=-20 \), at \( x = 8 \), \( f(8)=30 \). So the average rate of change is \( \frac{30 - (-20)}{8 - (-3)}=\frac{50}{11}\approx4.55 \). But let's confirm the formula. The average rate of change is the slope of the secant line between the two points \((-3,f(-3))\) and \((8,f(8))\). So yes, \( \frac{f(8)-f(-3)}{8-(-3)} \).

Wait, maybe I made a mistake in the y - values. Let's look at the graph again. The left part: when \( x=-3 \), the point is at \( y=-20 \) (since it's 2 units below 0, each unit 10? Wait, no, the y - axis has 0, 20, 40, - 20, - 40, etc. So each grid line is 10? Wait, the top of the graph has a peak at (0,40). Then at \( x=-3 \), the point is at \( y=-20 \), and at \( x = 8 \), the point is at \( y = 30 \). So the calculation is \( \frac{30-(-20)}{8 + 3}=\frac{50}{11}\approx4.55 \). But let's check the formula again. The average rate of change formula is correct: \( \frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b - a} \).

Answer:

\(\frac{50}{11}\) (or approximately \(4.55\))