We have this function :
$f(x) = \sqrt{1-x^2}$
$g(x) = \sqrt{x^2 +4}$
and I want to graph these functions using shifting (or transferring) graph of $\sqrt x$ , $x^2$ or any graph that can helps. (I don't want use point to point method for graphing plots) . Finally , I want to know how draw this function in general :
$h(x) = \sqrt{ax^2+bx+c}$.
Please help me!
$\sqrt{r^2-x^2}$ is the upper half of a circle with radius $r$. When you have a negative $x^2$ term, the result will look like an ellipse or a circle. When you have a positive $x^2$ term, the result will look like an absolute value function (it isn't though) with the bottom cut off if at any point the stuff in side the square root is negative or rounded if the stuff inside the square root is positive. It should be a hyperbola.
Finally, instead of writing the function as $\sqrt{ax^2+bx^2+c}$, write it more as $a\sqrt{(x-h)^2+k^2}$ or $a\sqrt{k^2-(x-h)^2}$, where $h$ represents the x-coordinate at which the function has its lowest value, $k$ represents its minimum or maximum, and $a$ represents a vertical scaling factor. In this form, you should be able to sketch it pretty easily.
It might help you to also look at this using conic sections, specifically ellipses and hyperbolas. The standard equation for an ellipse is $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ and the standard equation for a hyperbola is $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$, where $a$ and $b$ are scale parameters, and $(h,k)$ is the center.