Question

Find the average rate of change for the given function from x = -1 to x = 5.

Answer 1

@texaschic101

Answer 2

@sourwing

Answer 3

@kc_kennylau Do you know how to do this?

Answer 4

The average rate of change from point a to point b, is just the value of the function at point b - the value at point a, divided by the quantity (b-a).

Answer 5

So what do I divide?

Answer 6

Say you have a function \(f(x) = 3x+2\) (I'm not looking at your document — I don't open files that I have to download from unknown people)

To find the average rate of change of \(f(x)\) between \(x = a\) and \(x = b\), you evaluate this:

\[m = \frac{f(b)-f(a)}{b-a}\]

That gives you the slope of a line which connects the value of the function at points \(x = a\) and \(x = b\). That slope is the average rate of change, because that's what slope is!

Answer 7

Since you won't open the table the table says:
x y
-1 7
3 3
5 1

Answer 8

And its saying to "Find the average rate of change for the given function from x = -1 to x = 5."

Answer 9

So what numbers do i plug in..

Answer 10

\(y = f(x)\) so your table could just as well be labeled \[x\qquad f(x)\]

Answer 11

Find the value -1 in the x column, and find the value in the y column in the same row. Those are \(a\) and \(f(a)\) respectively.

Now find the value 5 in the x column, and the value in the y column of that row. Those are \(b\) and \(f(b)\) respectively.

Plug the 4 numbers into the formula. What do you get?

Answer 12

For giggles, we'll do the problem again with a formula: \(y = -x+6\) but let's get your answer to this first.

Answer 13

Still have no idea where to plug in what you're telling me to. :I I'm a really visual person :I so...I don't know where to plug in -1 but the y value next to it is 7. Still have no clue where to plug in 5 but the y value next to that is 1.

Answer 14

Would it be 7(-1)-5(1)/-1-5

Answer 15

Which would come to -12/-6 then reduce to 2 but that's not an option so where'd I go wrong

Answer 16

You're getting close!

Here's the table

x f(x)
-----------
-1 7
3 3
5 1

We want to find the average rate of change of f(x) (aka "the function") between x = -1 and x = 5.

We evaluate the function (read: look up the value in the table) at x=a=-1:

x = -1, f(-1) = 7

We evaluate the function at x = b = 5
x = 5, f(5) = 1

any question about that so far?

Answer 17

Okay understood

Answer 18

Now what

Answer 19

The starting point is (-1,7), or in my terminology, \((a, f(a))\)

The ending point is (5,1) or in my terminology, \((b, f(b))\)

We need to plug those into this formula:

\[m = \frac{f(b)-f(a)}{b-a}\]

Answer 20

That's what I thought I did but I clearly did it wrong

Answer 21

Try again. You put in a bunch of extra numbers. Very carefully match up the items I identified and write the formula

Answer 22

Here, I'll take you through it step by step:

What is \(a=\)

Answer 23

1?

Answer 24

Or no

Answer 25

I don't know

Answer 26

I told you the value in several different places...it is the x value for the starting point. We are finding the rate of change from x = -1 to x = 5.

Answer 27

I wrote a minute or so ago
"The starting point is (-1,7), or in my terminology, \((a,f(a))\)"

Match them up!

Answer 28

(-1,7)
(a, f(a))

Answer 29

Yeah sorry I have no clue I completely give up. Haha I'll just skip this question I don't have any more time

Answer 30

Thanks for trying to help though I guess

Answer 31

No, don't give up.

Answer 32

If I tell you there are two numbers, in order, (a,f(a)) and those two numbers are (-1,7), are you telling me you can't figure out the value of a?

Answer 33

I don't have time sorry I said -1 and you said no so

Answer 34

No, you said 1!

Answer 35

Oh my god -1 okay whatever

Answer 36

Typo

Answer 37

It makes a difference!

Answer 38

Ok

Answer 39

\[m=\frac{7-1}{5-(-1)}=\]

Answer 40

1

Answer 41

Cause 7-1=6 and 5-(-1) = +6

Answer 42

So 1.

Answer 43

That's right, but this time I made the mistake :-)

\[m = \frac{1-7}{5-(-1)}=\]

Answer 44

-6/6 = -1

Answer 45

There we go!

Answer 46

Here's a picture of what we did:

Answer 47

(-1,7) is the starting point, or (a,f(a)), way up in upper left corner.
(5,1) is the ending point, or (b,f(b)), bottom right corner

Answer 48

Here the points are joined and I put on grid lines to make it easy to see that for each step to the right along the x-axis, the y value goes down by 1. That's a slope of -1/1 = -1, just like we found.

Answer 49

If we'd wanted to find the average rate of change between another two points, say (1,5) and (3,3),

\[m = \frac{3-5}{3-1} = \frac{-2}{2} = -1\]Always will be the same if we are finding the average rate of change along a straight line like this