Question

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

Explanation:

Answer:

To estimate the average rate of change from \( x = 1 \) to \( x = 4 \), we first need to determine the values of the function at these points from the graph.

1. Find \( f(1) \): From the graph, at \( x = 1 \), the \( y \)-value (function value) appears to be \( 1 \) (assuming the grid lines correspond to integer values, and the curve passes through \( (1, 1) \)).
2. Find \( f(4) \): At \( x = 4 \), the graph has a vertex (minimum or maximum) at \( (4, -5) \) (estimating from the grid; the \( y \)-coordinate at \( x = 4 \) is \( -5 \)).

The formula for the average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Substituting \( a = 1 \), \( b = 4 \), \( f(1) = 1 \), and \( f(4) = -5 \):
\[
\text{Average Rate of Change} = \frac{-5 - 1}{4 - 1} = \frac{-6}{3} = -2.0
\]

(Note: If the graph’s coordinates differ slightly, adjust the \( y \)-values. For example, if \( f(1) \) is \( 2 \) and \( f(4) \) is \( -4 \), the calculation would be \( \frac{-4 - 2}{3} = -2.0 \), so the result remains consistent.)

\(\boldsymbol{-2.0}\)