Question

What is the average rate of change from x = 0 to x = 4?

0
1
4
8

Not sure if I'm correct

Answer 1

Don't have the function?

Answer 2

than you see the grapf from x=0 to x=2 so y decreasing from y=4 to y=-4 and from x=2 to x=4 so y increasing from y=-4 to y= 4

right @robtobey ?

Answer 3

so it's just x =2 then?

Answer 4

there is from x=2 to x=4 so y increasing from y=-4 to y=4

Answer 5

average rate of change of a function \(f(x)\) from \(x=a\) to \(x=b\) is simply\[\frac{f(b)-f(a)}{b-a}\]

Find the value of the function at the ending point. That's \(f(b)\). Find the value of the function at the starting point. That's \(f(a)\). Subtract. Divide by the ending value of \(x\) - the starting value of \(x\).

Average rate of change is simply the slope of the line going through the starting point and the ending point. rise/run

Answer 6

What do you mean by ending point?

Answer 7

and starting point?

Answer 8

you are finding the average rate of change from x = 0 to x = 4, right? The starting point is the value of y at x = 0, the ending point is the value of y at x = 4.

Answer 9

sorry, my line didn't quite line up, but I hope you get the idea

Answer 10

II don't know what f(b) would be 4, but f(a) would be 4 as well?

Answer 11

And if I used the formula you showed me it would be 0/4 which would just be 0?

Answer 12

right, @whpalmer4 ?

Answer 13

@Serenity74 ?

Answer 14

Is it 8 or 4 ?

Answer 15

yes. the average rate of change between \(x=0\) and \(x=4\) is \(0\) because the value of \(y\) is identical at those two points. It does not matter what happened in between. The average rate of change is just the total change (value at finish my value at start) divided by the number of units over which that change took place.

Viewed another way, the average rate of change is the number of units you must go up or down for each unit you move to the right to get from the starting point to the ending point. We start at \(y=4\) and we end at \(y=4\), so we do not have to go up or down at all. Our average rate of change is \(0\).

Note well that the average rate of change may be a very different value with a different starting and ending point! If we go from \(x=0\) to \(x=2\), \(y\) goes from \(y=4\) to \(y=-4\) for a change of \(-4 - 4 = -8\). Our average rate of change is \[\frac{f(b)-f(a)}{b-a}=\frac{(-4)-(4)}{2-0}=-4\]For every \(1\) unit we move to the right on the graph, we move \(4\) units down, when traveling along the line connecting those two points. Again, all that matters are the end points, not what happens in between them. Two functions with the same starting value and ending value over the interval of interest have the same average rate of change, even if one is a straight line between the two points, and the other one goes absolutely nuts.

Yet another way of thinking about it: You have 100 dollars in the bank, and get 20 dollars each week for your allowance. You don't spend any of it, just put it in the bank. Over the next two weeks, your bank balance increases by 2*20 = 40 dollars, so the average rate of change is \[\frac{140-100}{2-0}=20\]or an increase of 20 dollars per week. Your friend Tom also starts with 100 dollars, but he blows 50 dollars on flowers for his girlfriend on Valentine's Day, then wins 20 dollars playing poker with his friends and sells his old bicycle for 70, leaving him with \(100-50+20+70=140\) dollars. His average rate of change over the same period is also 20 dollars per week, but if you looked at a different interval, you might get a much different answer.