I'd like to introduce Taylor polynomials by generalizing the linear approximation of a function $f(x)$ to a quadratic approximation.
The linear approximation formula $L(x)=f(a)+f'(a)(x-a)$ of $f(x)$ near $a$ can be seen to be the point-slope formula of a line passing through the point $(a,f(a))$ with slope $f'(a)$.
Now, I'd like to derive the formula for a quadratic approximation of $f$ near $a$. Let this quadratic approximation be denoted by $T_2(x)$, and require that it satisfy the following $3$ conditions:
What is a nice way to show that $T_2(x)$ is what we know it to be without just writing it down and showing it has the desired properties? How can I motivate the coefficients? In other words, we don't really have a well-known ``point-slope-concavity" formula for parabolas that generalizes the point-slope formula we used to write down the linear approximation. I know I could start with a general form of $T_2(x)=Ax^2+Bx+C$, and use the above conditions to compute the coefficients, but for $a\ne 0$ this will be messy and unintuitive. If I could somehow motivate the switch to consider the form $T_2(x)=A(x-a)^2+B(x-a)+C$, that would be better, but I'm unsure how to convince the students that this is a natural form to consider.
The best I've been able to do is start off assuming that $T_2(x)$ is of the form $$T_2(x)=f(a)+f'(a)(x-a)+c(x-a)^2$$ for some constant $c$ but I think this is too much assuming too soon. Sure we can test that this form will satisfy $1$ and $2$ above, and that $c$ gives us the flexibility to specify $T_2''(a)$ so as to satisfy $3$, but this is still unmotivating to me. Why did I think this would work? It seems I've just written down what I know will work, and have only left the challenge of finding the right $c$ (which is just a computation and gives no intuition).
How can I motivate the form for $T_2(x)$ I've started with? Or is there another way to do this? I'd like to avoid viewing Taylor polynomials as truncated Taylor series, as I'm hoping to introduce Taylor polynomials before discussing infinite and power series.
Of course, I'm hoping that if I can intuitively motivate the derivation of $T_2(x)$, then a generalization to $T_n(x)$ will be clear.