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How to Estimate Average Rate of Change of a Function on Interval [2,5] from a Graph
Mathematics
Grade 11 (Senior High School)
Question Content
b. Estimate the average rate of change on the interval [2,5]
Correct Answer
-2
Detailed Solution Steps
1
Step 1: Use the average rate of change formula: $\\frac{f(b)-f(a)}{b-a}$ for interval $[2,5]$
2
Step 2: From the graph, identify the function values: at $x=2$, $f(2)=2$; at $x=5$, $f(5)=-4$
3
Step 3: Substitute into the formula: $\\frac{f(5)-f(2)}{5-2} = \\frac{-4 - 2}{3} = \\frac{-6}{3} = -2$
Knowledge Points Involved
1
Average Rate of Change
The average rate of change of a function $f(x)$ over the interval $[a,b]$ is calculated by $\\frac{f(b)-f(a)}{b-a}$. It represents the slope of the secant line connecting the points $(a,f(a))$ and $(b,f(b))$ on the function's graph, measuring the average rate at which the output changes relative to the input over the interval. A negative value indicates the function decreases on average over the interval.
2
Reading Function Values from a Graph
To find $f(x)$ for a given $x$, locate the $x$-value on the horizontal axis, move vertically to the graph of the function, then move horizontally to the vertical axis to read the corresponding $y$-value (the function output).
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