Determining the possible values of y for the graph y=x+2/3-x^2

[TFGordon]

Homework Statement

Determine the possible values that y can take for the graph y=x+2/3-x^2

Homework Equations

N/A

The Attempt at a Solution

y=k
k(3-x^2)=x+2
3k-kx^2=x+2
-kx^2+3k-x-2=0

I've tried factorising my k terms and much more faffing about to no avail...help please? :s

 

[micromass]

Hi TFGordon

I suppose you need to find the range of this function. This is very easy, you'll just need to solve the equation for x, and see where it is undefined.

Example, y=x². We need to solve this for x. We get $$x=\sqrt{y}$$ or $$x=-\sqrt{y}$$. In either case, this equation is undefined for y<0. Thus if we have $$y\geq 0$$, then there exists a corresponding y-value. Otherwise, such a y-value does not exist. Thus $$\mathbb{R}^+$$ is our range...

 

[hunt_mat]

Complete the square:

$$y=-\left( x+\frac{1}{2}\right)^{2}+\frac{11}{12}$$

What happens when x=1/2? Does this graph have a maximum/minimum?

 

[TFGordon]

I'm a further mathematics A level student, micromass. I appreciate your help but do you really think I would be attempting to to solve this equation if I was unaware that x^2=y works out as x=y^1/2? I know how graphical inequalities work pal, my problem is with the algebra, not the concept of an inequality.

Thankyou hunt mat! Just the ticket :) As it happens, the next part of the question is 'Find the co-ordinates of the stationary points of the curve' so yes, I imagine it does. Thankyou very much, I'd completely neglected to consider that approach :)

 

[micromass]

I'm a further mathematics A level student, micromass.

So? No need to get an attitude here...

I appreciate your help but do you really think I would be attempting to to solve this equation if I was unaware that x^2=y works out as x=y^1/2?

If you want better help, then you should have written more information in your attempt. You think it's easy to identify somebody's problems? I thought that you had problems with the general method. If you had problems with the algebra, then you should have written that.

I know how graphical inequalities work pal, my problem is with the algebra, not the concept of an inequality.

Very nice, but I don't quite see how graphical inequalities come into play here..

Anyway, let me give you another example: $$y=x^2+x$$. This corresponds to the quadratic equality $$x^2+x-y=0$$. So, the discriminant is $$D=1+4y$$ This is positive if $$y\geq -1/4$$. So the range is $$[-1/4,+\infty[$$.
With this example, you can easily calculate the range in your problem. It's the same thing really. So further mathematics A level student should have no problems with it...

 

[SammyS]

Do you mean: y=(x+2)/(3-x²) ?

Find the local min & max.