Is it possible to check if transformed function is correct using test points? Most of the (basic) math I've been doing in school has been easily checkable - it is immediately obvious if one had the right answer by just plugging it back in. Recently, doing function transformations, we were not taught a method to check if a transformed function was correct, and I am unable to figure it out myself.
Say we had an original function - $x^2-1%$ for example. I wanted to translate this function 2 units right, then reflect horizontally (over the y-axis). So I came up with the equation $(-x-2)^2-1$.
Plotting these two functions, with the original on the right:
http://i.stack.imgur.com/TKjiy.png (Image of the two graphs)
I can start at a point like $(0, -1)$, then add 2 units right, and reflect, and easily see that this transformation is correct. However, I would like to check algebrically if possible.
I've tried inputting the x coordinate in the transformed function and then trying to solve to see if I get the same y value, to no avail. I'm not sure how I can even guarantee a point exists on the original graph! 
Summary: Given an original function, and a transformed function, is it possible to check if the transformed function satisfies applied transformations?
 A: In your particular example, your function is a polynomial of degree $n$ transformed to another polynomial. Your function $-x^2 - 1$ is a polynomial of degree 2 (also known as a quadratic). Whenever that is the case, you can rely on the unisolvence theorem, the fact that a polynomial of order $n$ is uniquely determined by $n + 1$ points. So if you test $n + 1$ points and they go where you expected them to go, you can be sure that you have performed the correct transformation.
A: The answer to the question in the title is yes, it is possible to use test points.
The test point(s) should not be too simple, and two are better than one. Of course you have to carry out the transformations on the test points correctly. 
You can also do a backwards check. If for example translation to the right by $2$ followed by reflection in the $y$-axis gets you a certain result, then reflecting in the $y$-axis and translating left by $2$ should get you back to the original. Please note that going backwards involves undoing your transformations in the order opposite to the one that you used in the forward direction.
Here is an example with test points, using the curves that you discussed. So we start with $y=x^2-1$.  We move the curve $2$ units to the right. So $2$ plays the role that $0$ used to, and we get the curve $y=(x-2)^2-1$.  Then we reflect in the $y$-axis. Reflecting in the $y$-axis means that you replace $x$ everywhere by $-x$. Thus $y=(x-2)^2-1$ becomes $y=(-x-2)^2-1$.  This many minus signs is kind of ugly, and can lead to error. So since $-x-2=-(x+2)$, we end up with the nicer equation $y=(x+2)^2-1$.
Now let's check, by taking a point on the curve $y=x^2-1$. Let $x=3$. Then $3^2-1=8$, so our test point is $(3,8)$.
Move this $2$ units to the right. We get $(5,8)$. Reflect this new point across the $y$-axis. We get $(-5,8)$. Now we ask, is $(-5,8)$ on the curve $y=(x+2)^2-1$? Substitute $-5$ for $x$ in $(x+2)^2-1$. We get $(-5+2)^2-1=9-1=8$. Check done! 
