This is a follow-up and add-on to johnnyb's excellent and important answer.

One of the most important themes in Algebra 2 and higher-level classes is that of function transformation. This theme runs throughout virtually every topic in those courses. Consider the following function types:

• Linear functions: $y = A(x-x_0) + y_0$
• Quadratic functions: $y = A(x-x_0)^2 + y_0$
• Exponential functions: $y = Ab^{x-x_0} + y_0$
• Sinusoidal functions: $y = A\sin(x-x_0) + y_0$
• Logarithmic functions: $y = A\log_b (x-x_0) + y_0$

All of these functions have the same basic form: take a "toolkit" function ($y=x$, $y=x^2$, $y=b^x$, $y=\sin x$, or $y=\log_b x$) and perform (in sequence) the following transformations:

1. shift it horizontally
2. rescale it vertically
3. shift it vertically

Now you might have noticed that "rescale it horizontally" is missing from the list above. For linear and quadratic functions, there's a good reason for that: if $f(x)=x$ then $f(k\cdot x) = k\cdot f(x)$, which shows that a horizontal "compression" by a factor of $k$ is indistinguishable from a vertical "stretch" by the same factor; similarly if $g(x)=x^2$ then $g(k\cdot x) = k^2\cdot g(x)$, so a horizontal "compression" by a factor of $k$ is the same as a vertical "stretch" by a factor of $k^2$. (Incidentally, this is one of the reasons why when students are learning about quadratic functions in vertex form they often mis-recognize "make it taller" as "make it skinnier".) Even for exponential functions, a horizontal rescaling can be absorbed into a change of the base, so you don't (strictly speaking) need them. It's really only for trigonometric functions that the horizontal scale factor plays a role that can't be accounted for by one of the other parameters.

In high school, transformations are usually taught as a discrete topic, confined to an early chapter of Algebra 2 and only occasionally mentioned again, but I really think it is one of the most important throughlines of mathematics at the secondary and early postsecondary level.

In teaching university-level Calculus, for example, I will often begin the semester with a set of questions like this:

1. Find a formula for a linear function that passes through $(2,5)$ and $(9,15)$.
2. Find a formula for an exponential function that passes through $(2,5)$ and $(9,15)$.
3. Find a formula for a quadratic function that has its vertex at $(2,5)$ and passes through $(9,15)$.
4. Find a formula for a trigonometric function that has a minimum at $(2,5)$ and whose first maximum after that minimum is at $(9,15)$.

These are meant to be review problems, although students always struggle with questions 2-4. Needless to say, every student can answer question #1. Most will use point-slope form; a few will use slope-intercept form, and then rewrite their answer in point-slope form.

However, when we move on to #2, most students go through elaborate gyrations to find the values of $A$ and $b$ so that $y=A\cdot b^x$ passes through the given points. Then I show them that the function can be described as

When $x=2$ set $y=5$, and multiply $y$ by $3$ every time $x$ increases by $7$

and that this translates directly into the formula $$y=5\cdot 3^{(x-2)/7}$$

Students are usually astonished that it can be done that easily, at which point I remark that this is really the difference between slope-intercept form (which requires you to describe the line in terms of its $y$-intercept, regardless of whether that point is actually important for you) and point-slope form (which allows you to directly use the points you are given). We then spend the next couple of days approaching the rest of the problems using similar methods. This not only works well as a review of prerequisite material but also sets the stage for finding the equations of tangent lines, and (much later) Taylor series.

The point-slope form is (or has the potential to be) students' first encounter with transformational thinking and lays the groundwork for these more advanced topics. I think it is essential material for high school algebra.