Question

Two students in your class, Hunter and Maggie, are disputing a function. Hunter says that for the function, between x = -2 and x = 2, the average rate of change is 0. Maggie says that for the function, between x = -2 and x = 2, the graph goes up through a turning point, and then back down. Explain how Hunter and Maggie can both be correct, using complete sentences.

Answer 1

@SolomonZelman

Answer 2

we already did this one, didn't we?

Answer 3

uh no!

Answer 4

oh yeah we did sorry wrong one

Answer 5

it's alright

Answer 6

post the actual question (if any) below, in your next reply.

Answer 7

In your lab, a substance's temperature has been observed to follow the function T(x) = (x - 2)3 + 8. The turning point of the graph is where the substance changes from a solid to a liquid. Explain to your fellow scientists how to find the turning point of this function, using complete sentences.

Answer 8

okay, a turning point is a change either:
was decreasing, and now increasing
was increasing, and now decreasing

Answer 9

ok

Answer 10

the easiest thing would be to graph it, are you allowed to do this?

Answer 11

no its a written response

Answer 12

or maybe we can and then explain the graph

Answer 13

but i cant insert a graph

Answer 14

ok, sure.
In this case you will have only one turning point, it is the vertex.
in \(\large\color{black}{ S(x)=x^3 }\) it would be (0,0)
but your function is shifted:
\(\large\color{black}{ S(x)=x^3 ~~~~\Rightarrow~~~T(x)=(x\color{red}{-2})^3\color{blue}{+8} }\)

(shifted 2 units \(\large\color{black}{ \color{red}{\rm to~the~right} }\) and 8 units \(\large\color{black}{ \color{blue}{\rm up} }\) )

Answer 15

type if you want to say something, regardless of if I am typing or not.

Answer 16

ok i will

Answer 17

openstudy is loaded with viruses.. I disconnected 50 times just now

Answer 18

anyway, can you tell me where the turning point is?

Answer 19

what does that mean again?

Answer 20

is it -1,-1 and 1,1?

Answer 21

in a case y=x^3 it would be the origin, your function is shifted like this: (0+2 , 0+8 )

Answer 22

so what is the turning point?

Answer 23

I shifted the function 2 units right and 8 units up

Answer 24

oh smh i was thinking that i just dont like to be wrong

Answer 25

(2,8)

Answer 26

yes.

Answer 27

because it started from 0,0 right

Answer 28

n also (-2,-8)

Answer 29

Yes,

and there is a proof from calculus:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\(\large\color{black}{T(x)=(x-2)^3+8 }\)

take the derivative:
\(\large\color{black}{T'(x)=3(x-2)^2 }\)

set T'(x)=0
\(\large\color{black}{0=3(x-2)^2 }\)

\(\large\color{black}{0=(x-2)^2 }\)

\(\large\color{black}{0=x-2 }\)

\(\large\color{black}{2=x }\)

Answer 30

because its plus or minus right or wrong?

Answer 31

no, just and just 92,8)

Answer 32

oops, (2,8)

Answer 33

ok thanks

Answer 34

So, 2 is a critical number:
\(\large\color{black}{T(\color{red}{2})=(\color{red}{2}-2)^2+8 }\)
\(\large\color{black}{T(\color{red}{2})=8 }\)

Answer 35

yes i see it

Answer 36

yes, you welcome:)
(and there is no absolute max or min)

Answer 37

ok