The zoo of functions, and one grammar

A small zoo, not an infinite one

You have met functions one at a time across the last three modules: lines, parabolas, exponentials, logarithms, sine, cosine. It can feel like an endless bestiary.

It is not. Almost every function you will graph in this course is one of about ten basic shapes, the parent functions, possibly bent a little. Learn the ten on sight and you have the whole zoo.

Flip through the deck. For each card, notice the same five questions getting answered:

End behavior: where does the curve go as $x$ runs far left and far right?
Intercepts: where does it cross the axes?
Asymptotes: is there a line the curve hugs but never touches?
Periodicity: does the shape repeat?
Symmetry: is it a mirror image across the $y$ -axis (even), or does it spin onto itself through the origin (odd)?

Those five questions are the vocabulary. You do not memorize graphs; you describe them.

Read the degree

A polynomial’s degree is its highest power of $x$ , and the degree controls its end behavior.

What is the degree of $x^4 - 2x^2 + 1$ ?

Outside and inside

Here is the move that turns ten shapes into all of them.

Take any parent function $f$ . You can change its graph in exactly two places:

Outside the function: $a \cdot f(x) + v$ . This happens after $f$ has fired. It moves the output.
Inside the function: $f(k \cdot x - \text{something})$ . This happens before $f$ fires. It moves the input.

Outside changes are vertical, because the output is the $y$ -value. Inside changes are horizontal, because the input is the $x$ -value. That single split, outside versus inside, is the whole grammar.

The full template is:

g(x) = a \cdot f\big(k(x - h)\big) + v

Four knobs: $h$ slides it sideways, $v$ slides it up and down, $a$ stretches it vertically, $k$ squeezes it horizontally.

Turn the knobs

Drive the four knobs yourself. The widget keeps the faint parent curve drawn so you can always see what moved.

Start with the parent $x^2$ . Drag the shift handle: the parabola slides, shape untouched. Drag the vertical stretch handle: it gets taller or, past zero, flips upside down. Drag the horizontal handle: it pinches inward.

Watch the formula and the plain-English step list rewrite themselves as you drag. That synchronization, picture and symbols moving together, is the thing to internalize. The formula is not a separate fact to memorize; it is a readout of what your hands just did.

Slide it sideways

Consider $g(x) = (x-3)^2 + 5$ , the parent $x^2$ with its knobs set.

How many units to the right is this graph shifted?

Inside acts backward

One trap, and it catches everyone.

The graph of $f(x - 3)$ shifts right by 3, not left. The minus sign looks like it should pull the graph toward the negatives, but it does the opposite.

Here is why, with no hand-waving. The new graph at $x = 3$ shows the value $f(3 - 3) = f(0)$ . So whatever the parent did at its origin now happens at $x = 3$ . The origin moved right. Inside operations always run backward from what the sign suggests.

The same reversal hits the order of steps. Worked example: sketch $g(x) = -2(x-3)^2 + 5$ .

Inside, $x - 3$ : shift right 3.
Outside, the $-2$ : stretch vertically by 2, then reflect across the $x$ -axis.
Outside, the $+5$ : shift up 5.

Outside operations apply in the normal order of operations. Inside operations apply in reverse. Get inside-versus-outside right and the rest follows.

Period of a squeezed wave

The parent $\sin x$ repeats every $2\pi$ . The inside multiplier in $\sin(2x)$ squeezes it horizontally by a factor of 2.

What is the period of $\sin(2x)$ ? (As a decimal.)

The wave was using this grammar all along

Switch the widget’s parent to $\sin$ .

Look at what the knobs become. The vertical stretch $a$ is the amplitude you met in m3. The horizontal squeeze $k$ is the frequency. The vertical shift $v$ is the midline offset.

Module 3 taught you amplitude, frequency, and offset as trigonometry vocabulary. They were never trig-specific. They are just $a \cdot f(k x) + v$ with $f$ happening to be sine. One grammar, one widget, every parent function. The wave is not a special case; it is the same machine.

Find the midline

Consider $y = 3\sin(2x) - 1$ . The amplitude is 3, the period is $\pi$ , and there is an outside shift.

The wave oscillates around the line $y =$ what?

Where this shows up

A transformer is built from layers, and almost every layer is a transformation in exactly this sense. A learned weight matrix is, geometrically, a stretch and a reflection applied to its input vector. The positional encodings that tell the model where each token sits are shifted sine waves, $\sin$ with the $h$ and $k$ knobs turned.

You just learned the grammar those layers speak. Next we leave smooth curves for a moment and watch what happens when you add infinitely many numbers together.