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For slope, people sometimes use the description 'rise divided by run'. We know that the y axis describes rise, and the x axis describes run. So rise divided by run written numerically is . Now let's look at the line above. We can see that every time the y value goes up one unit, the x value goes right two units. This means that [tex]\frac{y}{x} --> \frac{y + 1}{x + 2}
This can be rewritten as 1/2, so the is 1/2. Hope this helped.
The average rate of change is also known as the slope. If we are looking to find the average rate of change between x = -1 and x = 2, in order to find the slope between those 2 x values, we need the y's that go along with them. Fill in -1 for x in the function and solve for y: . That means that the coordinate is (-1, 0). Fill in 2 for x in the function and solve for y:
. The coordinate there is (2, -3). Now we can set up the slope formula to find the average rate of change between those 2 points:
. Therefore, the average rate of change between x = -1 and x = 2 is -1