Question

Is the average slope between two points on the graph y=sqrt(x) always 0? Because if you have 3 and 5 on the y axis it must also be -3 and -5, therefore the two slopes are opposites and cancel to zero... right?

Answer 1

to be clear, the bigger question is how to determine average slope between points which are on a non-function line. Although please specificly answer this one. Also, would you rewrite x=y^3+8 as y=cbrt(x-8) right?

Answer 2

No, because if the 2 random points on y=sqrt(x) are (a,b) and (c,d)...
sqrt(a)=b --------> a=b^2
sqrt(c)=d --------> c=d^2
Slope between (a,b) and (c,d) is (d-b)/(c-a), or (d-b)/(d^2 - b^2).
(Factor and cancel out (d-b), assuming that (a,b) and (c,d) aren't the same point.)
(d-b)/((d-b)(d+b))=1/(d+b)

Answer 3

\[y = \sqrt{x}\]
plug in 4 for x
\[y = \sqrt{4} = 2\]
(4, 2)
plug in 9 for x
\[y = \sqrt{9} = 3\]
(9,3)

Average of two points:
(4 + 9)/2 , (2 + 3)/2
(13)/2 , (5)/2

Average of x coordinates : 6.5
Average of y coordinates: 2.5

Is that what you mean?

Answer 4

In conclusion, the slope between 2 points, (a.b), (c,d) that are on y=sqrt(x) is 1/(d+b), assuming that those 2 aren't the same points.
Now, 1/(d+b) can never be zero, so the average slope can never be zero.

Answer 5

For you second question, yes. x=y^3 + 8 can be rewritten as y=cbrt(x-8) without any restrictions.

Answer 6

but science, if you pick x=4 then y is 2 AND -2 so you have two slopes to deal with

Answer 7

sqrt(4) is not -2 because square root of any positive real number gives you positive real number.

Answer 8

If you draw the graph, you'll notice that the range is restricted to [0,infinity)

Answer 9

Also, by the definition of a function, if 2 different x-values have the same y-value, that is not function.

Answer 10

So, no function can have (2,4) and (-2,4) as its points.

Answer 11

y=sqrt(x) is a function, so if x=4, y=2.

Answer 12

no it isn't xD -2*(-2)=4 and 2*2=4

Answer 13

So, -2*-2=2*2=4 ----> sqrt((-2)^2)=sqrt(4) ------> -2=2
Are you saying that?

Answer 14

Try to figure out what's wrong...

Answer 15

no lol sqrt((-2)^2) is sqrt(-2*(-2)) which is positive 4
so concluding -2=2 isn't correct. You would conclude with -2=-2 and 2=2

Answer 16

There is a real number property, which states that sqrt(a^2)=a.
If you substitute -2 for a, what happens?

Answer 17

Are you denying that a negative number times a negative number is a positive number? Sorry, but thats fundemental. There are very few people in this world who would agree that the graph y=sqrt(x) only is in quadrant 1. And that isn't a real number property its a natural number property.

Answer 18

I'm not denying that.
If a=2, sqrt(-2*-2)=-2 -------> sqrt(4)=-2 --------> 2=-2
What do you think is wrong here?

Answer 19

I hope this a friendly argument, math isn't something to get mad about :)

Answer 20

Don't worry.
This won't get serious

Answer 21

Let's start over.

Answer 22

lets go back to the original question

Answer 23

I have checked with other people and so far i have gotten two answers. The average slope between two points on a non-function graph like y=sqrt(x) is undefined, or it's always 0. I think saying that it is zero makes more sense because the slope of the part in the 4th quadrant is the exact opposite of that in the first.

Answer 24

First thing first, y=sqrt(x) is a function.

Answer 25

It passes the vertical line test.

Answer 26

y=sqrt(x) is only defined at the 1st quadrant and the origin.

Answer 27

no its not. it fails the vertical line test.

Answer 28

the domain is restricted to [0,infinity), right?

Answer 29

yes, x must be equal to or greater than 0.

Answer 30

otherwise you have a square root of a negative number which is on the complex plane, not the real plane.

Answer 31

Great.

Answer 32

Now, when x=0, what's y?

Answer 33

sqrt(0)=0 of course

Answer 34

but the square root of, lets take 4 for example, is \[\pm2\]

Answer 35

Why?

Answer 36

because @science0229
(-2)^2 = 4
and
(2)^2 = 4

Answer 37

I didn't ask that because I didn't know it.

Answer 38

Squar root of any positive real number ALWAYS give you positive answer. NEVER negative.

Answer 39

For example, sqrt(4) is not -2

Answer 40

that isn't the ONLY solution, but it is ONE of the solutions. Definition of "square root of x" is a number which when multiplied by itself gives the product of x. -2*(-2)=4 so one solution of sqrt(4) is -2.

Answer 41

It could be though, @science0229

Answer 42

Did you forget the restriction for square roots??????

Answer 43

in school, the answer is usually a positive number, unless they specifically ask for the negative root as well

Answer 44

That is specifically only for solving equations.

Answer 45

if it can be used in an equation, its true xD

Answer 46

Solving equations and taking a square root of a number is not the same thing...

Answer 47

If -2 wasn't a value for the square root of 4, than it looks like you have to correct just about every single math book ever written. Good luck with that.

Answer 48

No.

Answer 49

-2 is a solution of the equation x^2=4
-2 is not the answer for sqrt(4)

Answer 50

the reason that -2 is a solution of the equation x^2=4 is that negative number squared is always positive.
the reason that -2 is not sqrt(4) is that square root of a positive number can NEVER be a negative number.

Answer 51

you mean the square root of a negative can never be a positive number. that would be imaginary.

Answer 52

No. Sqrt(4) is 2, which is positive number.
It is not -2 because it is a negative number.

Answer 53

All of us are us are using the fact that sqrt(a^2)=a

Answer 54

You're repeating the same thing over and over, whats your PROOF. Find a definition that actually says negative numbers can't be solutions to the square roots of postives

Answer 55

But, there is a restriction that a>=0

Answer 56

where does anyone say that.

Answer 57

just look up "definition of square root" on google. It even says at the bottom "7 is A square root of 49" implying there are MORE THAN ONE. not 7 is THE square root of 49, 7 is A square root of 49.

Answer 58

There is a thing called square root, and there is a thing called principal and negative square root.

Answer 59

Square root always gives you POSITIVE answer.

Answer 60

Principal root always gives you POSITIVE AND NEGATIVE answers.

Answer 61

Negative root always gives you NEGATIVE answer.