Can all affine cyphers be expressed with this formula There are $26!$ permutations of the English alphabet.
In Modern Cryptography textbook, William Easttom states that

the basic formula for any affine cipher is $ax + b \equiv \pmod{26}$.

I'm wondering whether this formula can express every possible permutation. It does not seem obvious that for every permutation $\sigma$ there exist $a$ and $b$ capable of expressing it. If this statement is true, what approach can be used to find $a$ and $b$?
$a\cdot x +b $ means affine  not permutation. And $a\cdot x +b \bmod 26$ is modular affine transformation.
$a\cdot x +b \bmod 26$ can have at most $26\cdot 26$  affine transformations some of which have no inverse and therefore without an inverse an affine transformation is not suitable for encryption where it is already not close to the modern encryption methods.
To have an inverse one must have $\gcd(a,26)=1$. This means $a$ cannot be divisible by 2 or 13. To find the actual number we can use Euler's Totient Function that counts the positive integers up to a given integer $n$ that are relatively prime to $n$. Then $$\phi(26) = (13-1)\cdot (2-1) = 13$$ So there are actually $26\cdot 12 =312$ possible affine transformations.
In any case, we have $26! = 403291461126605635584000000$ permutations and $312$ affine transformations, therefore the affine transformation cannot cover all possible permutations. The reverse is correct; the permutations can represent the invertible affine transformations.