Example 12 A student, in order to plot the graph of the function $y=$ $a x^{2}+b x+c(a \neq 0)$, took 7 values of the independent variable: $x_{1}<x_{2}<\cdots<x_{7}$, and $x_{2}-x_{1}=x_{3}-x_{2}=\cdots$ $=x_{7}-x_{6}$, and calculated the corresponding $y$ values, listing them in Table 1.

But due to carelessness, one of the $y$ values was calculated incorrectly. Please point out which value is incorrect? What is the correct value? And explain the reason.
(2003, Shanghai Junior High School Mathematics Competition)
Explanation: Let $x_{2}-x_{1}=x_{3}-x_{2}=\cdots=x_{7}-x_{6}$ $=d$, and the function value corresponding to $x_{i}$ is $y_{i}$. Then
$$
\begin{array}{l}
\Delta_{k}=y_{k+1}-y_{k} \\
=\left(a x_{k+1}^{2}+b x_{k+1}+c\right)-\left(a x_{k}^{2}+b x_{k}+c\right) \\
=a\left[\left(x_{k}+d\right)^{2}-x_{k}^{2}\right]+b\left[\left(x_{k}+d\right)-x_{k}\right] \\
=2 a d x_{k}+\left(a d^{2}+b d\right) .
\end{array}
$$

Therefore, $\Delta_{k+1}-\Delta_{k}$
$$
=2 a d\left(x_{k+1}-x_{k}\right)=2 a d^{2} \text { (constant). }
$$

From the given data
$$
\begin{array}{llllllllll} 
& y_{k} & : & 51 & 107 & 185 & 285 & 407 & 549 & 717 \\
\text { we get } & \Delta_{k} & : & & 50 & 78 & 100 & 122 & 142 & 168 \\
\Delta_{k+1}-\Delta_{k} & : & & 22 & 22 & 22 & 20 & 26
\end{array}
$$

From this, we can see that 549 is the incorrect $y$ value, and its correct value should be 551.

Note: A quadratic function is essentially a second-order arithmetic sequence with the independent variable taking real numbers. This problem is designed based on such a background. The key to solving the problem is to grasp the essential characteristics through the phenomena.