A polynomial curve have turning points at x=3 and x=-2. It cut's the y axis at 10. Find a possible equation of the curve (demonstrating how you did so).

Guest Jan 24, 2015

#3**+5 **

And yet, here again is another possibility....if you have had calculus, we know that the derivative = 0 where the turning points occur,

Therefore, let the derivative of the function be (x -3) (x +2) = x^2 - x - 6.

Then, again.... this may only make sense if you have had calculus.......our basic function would be:

F(x) = (1/3)x^3 - (1/2)x^2 - 6x

This function will have a y intercept of 0, so we just need to append a "10" at the end to give us a y intercept of 10.

So....our "translated" function would be...

F(x) = (1/3)x^3 - (1/2)x^2 - 6x + 10

Here's the graph......https://www.desmos.com/calculator/jdknoo63eb

CPhill
Jan 24, 2015

#1**+5 **

Here's one possibility........https://www.desmos.com/calculator/o9djnpu1cs

I know that roots with an even multiplicity will"kiss" the x axis........so.....allI had to do was construct the polynomial y = (x-3)^2(x+2)^2..and this polynomial will have "turning points" at (0,3) and (0,-2).

It will have a y intercept at (0,36), so all I had to do was add the -26 at the end, and this shifts the polynomial "down" by 26 units so that the "new" y intercept is (0,10). This does not affect the values of the turning points.

Of course, this polynomial has one more turning point when x =1/2, but, as far as I know, you didn't say it couldn't have* more* than two turning points...

CPhill
Jan 24, 2015

#3**+5 **

Best Answer

And yet, here again is another possibility....if you have had calculus, we know that the derivative = 0 where the turning points occur,

Therefore, let the derivative of the function be (x -3) (x +2) = x^2 - x - 6.

Then, again.... this may only make sense if you have had calculus.......our basic function would be:

F(x) = (1/3)x^3 - (1/2)x^2 - 6x

This function will have a y intercept of 0, so we just need to append a "10" at the end to give us a y intercept of 10.

So....our "translated" function would be...

F(x) = (1/3)x^3 - (1/2)x^2 - 6x + 10

Here's the graph......https://www.desmos.com/calculator/jdknoo63eb

CPhill
Jan 24, 2015