linear function transformations explanation

Question

i am new to algebra and trying to self learn mathematics
i got a book and search google to understand function transformations
i find the following strange:
$$y = f(x)$$ It can be written in the format shown to the below.
$$\frac{y-d}{a} = f\left(\frac{x-c}{b}\right)$$

"here $y$ is divided by a which is the vertical multiplier" "here how the concept of multiplying by $a$ is written as dividing by $a$ "

and then in this format:
$$y = a \cdot f\left(\frac{x-c}{b}\right) + d$$ "here $x$ is divided by $b$ which is the horizontal multiplier"

and then in this format:
$$y = a \cdot f [ b (x-c) ] + d$$ "here $x$ is multiplied by $b$ which is the horizontal multiplier">>>>so how the divide by $b$ is the same as multiply by $b$???

thanks to Alexvong for his reply :
i found this proof
and wish to comment on it:

Theorem.
If the graph of $y = f (x)$ is translated $a$ units horizontally and $b$ units vertically, then the equation of the translated graph is

$$y − b = f(x − a)$$

For in a translation, every point on the graph moves in the same manner. Let $(x_1, y_1)$, then, be the coördinates of any point on the graph of $y = f (x)$, so that

$$y_1 = f (x_1)$$

And let us translate the graph $a$ units horizontally and $b$ units vertically, so that $x_1$ goes to the point

$$x_1 + a$$

and $y_1$ goes to the point

$$y_1 + b$$

If $a$ is a positive number, then that point will be to the right of $x_1$, while if $a$ is negative, it will be to the left. Similarly, if $b$ is a positive number, then $y_1 + b$ will be above $y_1$, while if $b$ is negative, it will be below.

Now, what will be the equation of the translated graph, such that when the value of $x$ in the equation is $x_1 + a$, the value of $y$ will be $y_1 + b$?

We say that the following is the equation:

$$y − b = f(x − a)$$

For, when $x = x_1 + a$:

$$y − b = f(x_1 + a − a) = f(x_1) = y_1$$

$$y = y_1 + b$$

And $(x_1, y_1)$ is any point on the graph of $y = f (x)$.
Therefore the equation of the translated graph is

$$y − b = f(x − a)$$

Which is what we wanted to prove.

Answer 1

Here the f( (x-c) / b ) has reverse meaning as following:
X divided by b means that the standard graph y=f(x) will be stretched horizontally because at y = 4 , x will equal 8/2=4 so x = 8
and the graph stretched “here b=2”>>>>>>so division by b means horizontal stretch” reverse meaning”

And when we move b from denominator (division by b) to nominator (multiplication by b) ,
we find that x multiplied by b means that the standard graph y=f(x) will be compressed horizontally because at y=4 , x will equal 2*2=4 so x =2 and the graph compressed horizontally “here b=2” so multiplication by b means horizontal compression “reverse meaning still”

so the f( (x-c) / b ) is equal in meaning to f [ b (x-c) ] >>>but using division to lead to stretch in first and using multiplication to lead to compression in second

the reverse meaning meant here is that division should lead to compression ”approaching y axis with lower x”
and multiplication should lead to stretching “away from y axis with higher x”

Answer 2

Firstly, given $y = f(x)$, after applying the transformations we have learned (scaling, shifting and reflection), we have $$y = a' \cdot f[b'(x - c')] + d'$$ for some $a', b', c', d'$. Note that $a', b', c', d'$ are just symbols representing real number, there are nothing mysterious about them.

Suppose the vertical scaling factor $a' \ne 0$ and horizontal scaling factor $b' \ne 0$. Then we claim $$y = a' \cdot f[b'(x - c')] + d'$$ can also be written in the form $$\frac{y - d}{a} = f\left(\frac{x - c}{b}\right)$$ for some $a, b, c, d$.

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Proof

By assumption, we have $$y = a' \cdot f[b'(x - c')] + d'$$ Let $a = a', c = c'$ and $d = d'$. After substitution we have $$y = a \cdot f[b'(x - c)] + d$$ Since by assumption $b' \ne 0$, we can let $b = \frac{1}{b'}$. After substitution we have $$y = a \cdot f\left(\frac{x - c}{b}\right) + d$$ Minus both sides by $d$ gives $$y - d= a \cdot f\left(\frac{x - c}{b}\right)$$ By assumption, $a = a' \ne 0$, so we can divide both sides by $a$ $$\frac{y - d}{a} = f\left(\frac{x - c}{b}\right)$$ which is what we need to show.