Ignoring the fact that you need the function's values actually at the particular point in question, you do in fact need to know something else: you need to know that the function in question has sufficient regularity (enough smoothness) to allow an application of Taylor's theorem.
http://www.corestandards.org/Math/Content/HSN/VM/
CCSS.MATH.CONTENT.HSN.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
CCSS.MATH.CONTENT.HSN.VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
Math education gets a bad rap in the USA, because anything that students don't remember learning, is claimed to have never have been taught. And that is just not true. Math teachers aren't as dumb as people pretend they are.
I have seen the standards for high school math teachers and trust me, they are low. Most of my classmates in undergrad (which was supposed to be an excellent program for math teachers) are now high school teachers. Their complaints about how much they hate basic linear algebra and "just want to be done with it so they can go teach high school" are still ringing in my ears.
For linear transformations, one possible avenue is start with the notion of derivative. If we take a real-valued function, its derivative at a particular point is just a single number, which represents the function's rate of change. But what if we have a function that accepts, say, two real numbers and outputs three? It turns out that the natural generalization of "derivative" to such functions is a rectangular array of numbers:
If we know these numbers (and nothing else), we can linearly approximate the values of a function near a particular point, with at most quadratic error.Now let's say we have two functions. The first one takes two numbers and outputs three, and the second one takes three numbers and outputs four. If we compose them together, can we find the derivative of the composite from the two simpler derivatives by some kind of chain rule, like the one we have for ordinary real-valued functions? It turns out that yes, we can, if we replace the product of two numbers with the product of two matrices (defined in a particular way).
Now it's easy to explain what linear transformations are. They are just multidimensional functions whose derivative (matrix) is the same at every point. They are just like one-dimensional linear functions, whose derivative (number) is the same at every point. (For convenience, people also say that every linear transformation must take the point (0,0,...) to the point (0,0,...), so that matrices correspond one-to-one to linear transformations and vice versa.)
If you want to work with linear transformations effortlessly, there's a lot more intuition to develop, but this should serve for the basics.