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How to Estimate Average Rate of Change of a Function on Interval [1,4] from a Graph
Mathematics
Grade 11 (Senior High School)
Question Content
a. Estimate the average rate of change on the interval [1,4]
Correct Answer
1
Detailed Solution Steps
1
Step 1: Recall the formula for average rate of change on an interval [a,b]: $\\frac{f(b)-f(a)}{b-a}$
2
Step 2: From the graph, identify the function values: at $x=1$, $f(1)=-4$; at $x=4$, $f(4)=-1$
3
Step 3: Substitute into the formula: $\\frac{f(4)-f(1)}{4-1} = \\frac{-1 - (-4)}{3} = \\frac{3}{3} = 1$
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.
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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