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The gradient of a straight line graph is calculated as:
for two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ on the graph.
We can now define the average gradient between two points even if they are defined by a function which is not a straight line, $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ as:
This is the same as [link] .
Fill in the table by calculating the average gradient over the indicated intervals for the function $f\left(x\right)=2x-2$ . Note that ( ${x}_{1}$ ; ${y}_{1}$ ) is the co-ordinates of the first point and ( ${x}_{2}$ ; ${y}_{2}$ ) is the co-ordinates of the second point. So for AB, ( ${x}_{1}$ ; ${y}_{1}$ ) is the co-ordinates of point A and ( ${x}_{2}$ ; ${y}_{2}$ ) is the co-ordinates of point B.
${x}_{1}$ | ${x}_{2}$ | ${y}_{1}$ | ${y}_{2}$ | $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ | |
A-B | |||||
A-C | |||||
B-C |
What do you notice about the gradients over each interval?
The average gradient of a straight-line function is the same over any two intervals on the function.
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