Question 19 8 7 6 5 4 3 2 1 -1 1 2 3 4 5 Based on the graph above, estimate to one decimal place the average rate of change from
to
The Correct Answer and Explanation is:
Answer: -1.0
Explanation:
The average rate of change of a function over a specific interval represents the slope of the secant line connecting the two endpoints of that interval on the graph. The formula to calculate the average rate of change of a function, let’s call it f(x), from x = a to x = b is:
Average Rate of Change = (f(b) – f(a)) / (b – a)
In this problem, we are asked to find the average rate of change from x = 1 to x = 3. Therefore, a = 1 and b = 3.
Step 1: Find the function values (y-values) from the graph.
First, we need to determine the y-values of the function at x = 1 and x = 3 by reading the provided graph.
- Find f(1): Locate x = 1 on the horizontal axis (the x-axis). Move vertically up from this point until you intersect the curve. From this intersection point, move horizontally to the left to read the value on the vertical axis (the y-axis). The graph clearly passes through the point (1, 5). Therefore, f(1) = 5.
- Find f(3): Similarly, locate x = 3 on the horizontal axis. Move vertically up to the curve. From that point on the curve, move horizontally to the left to read the corresponding y-value. The graph passes through the point (3, 3). Therefore, f(3) = 3.
Step 2: Apply the average rate of change formula.
Now, we substitute the values we found into the formula:
- a = 1, f(a) = f(1) = 5
- b = 3, f(b) = f(3) = 3
Average Rate of Change = (f(3) – f(1)) / (3 – 1)
Average Rate of Change = (3 – 5) / (3 – 1)
Average Rate of Change = -2 / 2
Average Rate of Change = -1
Step 3: Format the answer.
The question asks for the answer to be estimated to one decimal place. To express -1 with one decimal place, we write it as -1.0.
This result means that, on average, between x = 1 and x = 3, the function’s value decreases by 1 unit for every 1 unit increase in x
