Question

3.3: homework assignment score: 11.5/14 answered: 12/14 question 2 based on the graph above, estimate (to one decimal place) the average rate of change from x = 1 to x = 4. question help: video

Explanation:

Answer:

To find the average rate of change from \( x = 1 \) to \( x = 4 \), we use the formula for average rate of change: \( \frac{f(4) - f(1)}{4 - 1} \).

1. Estimate \( f(1) \): From the graph, at \( x = 1 \), the \( y \)-value ( \( f(1) \)) appears to be 4 (since the graph passes through or near \( (1, 4) \)).
2. Estimate \( f(4) \): From the graph, at \( x = 4 \), the \( y \)-value ( \( f(4) \)) appears to be 7 (since the graph has a peak near \( (4, 7) \)).

Now, calculate the average rate of change:
\[
\frac{f(4) - f(1)}{4 - 1} = \frac{7 - 4}{3} = \frac{3}{3} = 1.0
\]

Wait, maybe my initial estimates are off. Let's re-examine the graph. At \( x = 1 \), the graph is at a peak? Wait, no, looking at the grid: the vertical lines are at \( x = -1, 0, 1, 2, 3, 4, 5 \), and horizontal lines at \( y = 1, 2, 3, 4, 5, 6, 7, 8 \).

At \( x = 1 \): The graph is at a peak? Wait, no, the first peak is left of \( x = 0 \), then a valley at \( x \approx 0.5 \) ( \( y \approx 1 \) ), then a peak at \( x \approx 1.5 \) ( \( y \approx 4 \) ), then a valley at \( x \approx 2.5 \) ( \( y \approx 1 \) ), then a peak at \( x \approx 3.5 \) ( \( y \approx 7 \) ), then a valley at \( x \approx 4.5 \) ( \( y \approx 4 \) ), then rising.

Wait, maybe \( f(1) \) is 4 (at \( x = 1 \), the graph is at the peak around \( x = 1.5 \), but maybe \( x = 1 \) is at \( y = 4 \)). At \( x = 4 \), the graph is at the peak around \( x = 3.5 \) to \( 4 \), so \( y \approx 7 \). Then the change in \( y \) is \( 7 - 4 = 3 \), change in \( x \) is \( 4 - 1 = 3 \), so \( 3/3 = 1.0 \).

Alternatively, maybe \( f(1) = 4 \) and \( f(4) = 7 \), so average rate of change is \( (7 - 4)/(4 - 1) = 1.0 \).

So the answer is \( \boxed{1.0} \).