Question

Write a polynomial that would have 3 turning points, shifted to the right 7 units and be reflected over the x-axis.

Answer 1

a = -1 reflection
shift to the right x -7
3 turning points highest power in polynomial 3
f(x) = -(x-7)^3

Answer 2

this graph only shows 2 turns when plugged into the graphing calculator.
therefore not sure

Answer 3

@DanJS is this incomplete? @sleepyjess ?

Answer 4

a polynomial of 4th degree will have 3 or less turning points

Answer 5

where there is a local max or min

Answer 6

did you see what this graph look like?

Answer 7

ie - x^4 - 5x^2

Answer 8

would it be Ok to just add a factor x?

Answer 9

normally take the 2nd derivative to find the max or min

Answer 10

so what would be appropriate for this? something we make up or is it specific with the information given?

Answer 11

hmm, try factors i guess.

y = -(x-7)(x-3)(x+3)

Answer 12

that is a 3rd degree derivative

Answer 13

this worked for me according to the graph -x^2(x-7)^2

Answer 14

y = (x+3)(x-3)(x+7)(x-7) and let x=x-7 for the shift

y = -((x-7)+7)((x-7)-7)*((x-7)+3)*((x-7)-3)

Answer 15

the one above max(0,0) min (3.5, -150) max (7,0)

Answer 16

where did you get +/-3 to use as factor?

Answer 17

try

y = -(x-8)^2 * (x-6)^2

Answer 18

or

Answer 19

https://www.desmos.com/calculator

Answer 20

y = -(x-5)^2*(x-9)^2

Answer 21

y = -(x - 4)^2*(x-10)^2

Answer 22

notice the pattern

Answer 23

@jazzyrocroyal any ideas now?

Answer 24

-(x-3)^2(x-11)^2

Answer 25

so on...

Answer 26

this is just from me playing around. lol there may be a formal way to do that that is slipping my mind

Answer 27

@DanJS thanks for your help.

@jazzyrocroyal still nothing to say?
@Directrix hi just in time to close this out for us

Answer 28

welcome,

i would just go with y = -(x-8)^2 * (x-6)^2

Answer 29

If you have a -x^4 and a +x^2 term it works

-x^4 + x^2

-(x-7)^4 + (x-7)^2

Answer 30

that seems to be the trend with them, in general a 4th order polynomial will have 3 OR LESS turns, to have 3, looks like you need Opposite signs on the 4th order and 2nd order terms

Answer 31

and to have one opening down< you want a negative on the 4th order term, so you need a positive 2nd order term

Answer 32

exploring some more 8^2 +6^2 = 100
(8^2*6^2 )1/2 = 48 100 + 48 = 148 for y then only use the negative
close enough to -150
(8*6)^1/2 then divide2 = 3.4 no use 7/2 = 3.5 the x coordinate for the vertex

Answer 33

just a sample? this is same as @DanJS basically in regards to the choosing of exponents

Answer 34

will it satisfy the requirement for this or just a 3-point graph?

Answer 35

I looked. that's just a sample.

Answer 36

repost
this worked for me according to the graph -x^2(x-7)^2

Answer 37

did you look at the graph?

Answer 38

Think you just need to alternate the signs on each decreasing power of x, with the first being -x^4 so it points down, then replacing x by(x-7) for the shift

Answer 39

@jazzyrocroyal feel free to join in especially since this is your post

Answer 40

all even powers will be negative, all odd powers positive coefficients

Answer 41

does this have something to do with Pascal's triangle, i vaguely remember doing that

Answer 42

ah, got to run for the time, good luck ya'll

Answer 43

@DanJS execellent job thanks