Answer: The average rate of change for both is -2
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Explanation:
The x interval [0,3] is the same as writing [tex]0 \le x \le 3[/tex]
It starts at x = 0 and ends at x = 3.
The graph shows that x = 0 leads to y = 3. So we have the point (0,3) on the parabola. We also have the point (3,-3) on the parabola.
Let's find the slope of the line through these endpoints.
[tex](x_1,y_1) = (0,3) \text{ and } (x_2,y_2) = (3,-3)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{-3 - 3}{3 - 0}\\\\m = \frac{-6}{3}\\\\m = -2\\\\[/tex]
The slope is -2. This is the average rate of change from x = 0 to x = 3.
This is because:
slope = rise/run = (change in y)/(change in x) = average rate of change.
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Now let's find the slope for the table.
Focus on the rows for x = 0 and x = 3. They lead to f(x) = 10 and f(x) = 4 respectively.
We have (0,10) and (3,4) as our two points this time.
[tex](x_1,y_1) = (0,10) \text{ and } (x_2,y_2) = (3,4)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{4 - 10}{3 - 0}\\\\m = \frac{-6}{3}\\\\m = -2\\\\[/tex]
We get the same slope as before, so we have the same rate of change.
Notice the change in y (-6) is the same as before. So we could pick any two y values we want as long as there's a gap of 6 between them, and the second y value is smaller than the first.
