what is the domain, range, intercepts, symmetries, asymptotes, and graph of the
y^2 = x (x - 1) (x - 2)
i just have a very limited brain. need your guide and assistance guys. ( Sir ) God Bless
thanks
I hope this doesn't come too late ...
1. Domain: Since $\displaystyle y^2 \ge 0$ the RHS of the equation must be greater or equal zero too.
$\displaystyle x (x - 1) (x - 2) \ge 0~\implies~ x \ge 0$ If x < 0 then the other 2 factors are also negative which yields a negative result.
You now have to examine under the condition that $\displaystyle x \ge 0$
$\displaystyle (x - 1) (x - 2) \ge 0~\implies~ \underbrace{x-1\ge 0 \wedge x-2 \ge 0}_{x\ge 2}~\vee~ \underbrace{x-1 \le 0 \wedge x-2 \le 0}_{0 \le x \le 1}$
2. Range: In general $\displaystyle y \in \mathbb{R}$. But since the domain is split into 2 seperated intervals you have to find (especially for the 1st interval) any existing maximum values. Use calculus.
3. Intercepts:
x-intercepts occur if y = 0. Since you have the term of y^2 in factored form the x-intercepts are easily to determine.
y-intercepts occur if x = 0. So one x-intercept coincide with the y-intercept.
4. Symmetries: What do you know about graphs whose one variable is squared?
5. Asymptotes: $\displaystyle a(x)=\lim_{x \to \infty} y$
You have to consider to different equations of y. (One asymptote is drawn in blue)