possible equation

red and white kop!

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Jun 15, 2009
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im supposed to find a possible equation for this graph; i know that the equation must be degree 6 cos it has five turning points but otherwise i have on idea where to start, all of my attempts on graphic calculator failed miserably.
any hints?
 

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Hello, red and white kop!


I'll assume the graph symmetric to the y\displaystyle y-axis.

Choose six x\displaystyle x-intercepts, say: .-3,-2,-1,1,2,3\displaystyle \text{-}3, \text{-}2, \text{-}1, 1, 2, 3

Then the function is of the form: .f(x)=a(x+3)(x+2)(x+1)(x1)(x2)(x3)\displaystyle f(x) \:=\:a(x+3)(x+2)(x+1)(x-1)(x-2)(x-3)

Since the graph "opens downward", a\displaystyle a is negative.


One possible function is: .f(x)  =  (x+3)(x+3)(x+1)(x1)(x2)(x3)\displaystyle f(x) \;=\;-(x+3)(x+3)(x+1)(x-1)(x-2)(x-3)

. . . which simplifies to: .f(x)  =  x6+14x449x2+36\displaystyle f(x) \;=\;-x^6 + 14x^4 - 49x^2 + 36

 
Hello again, red and white kop!

but doesnt this make the y-values way too high?

There is no scale on your diagram.
. . How can the y-values be "too high"?


My graph has:   {Maxima:(±7,36)(0,36)Minima:(±73,-40027)(±1.53,14.81)}\displaystyle \text{My graph has: }\;\begin{Bmatrix}\text{Maxima:} & \left(\pm\sqrt{7},36\right) \\ & (0,36) \\ \\ \text{Minima:} & \left(\pm\sqrt{\frac{7}{3}}, \text{-}\frac{400}{27}\right) & \approx & (\pm1.53, -14.81) \end{Bmatrix}



 
i guess you're right, but still is there any way to decrease the y-values without tinkering with the x-intercepts?
i guess not
 
If you want to decrease the y values but leave the x intercepts alone, simply change the coefficient. Notice the 'a' Soroban has in his first post. Make that something besides -1 to change the heught of the y's


   change this value. Soroban used -1\displaystyle \;\ \text{change this value. Soroban used -1}
\displaystyle \downarrow
a(x+3)(x+2)(x+1)(x1)(x2)(x3)\displaystyle \fbox{a}(x+3)(x+2)(x+1)(x-1)(x-2)(x-3)
 
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