SL 2.11—Transformation of functions

When you have a function $f(x)$, you can create new functions by shifting, flipping, or stretching its graph. These operations are called transformations, and they all follow predictable rules , meaning once you learn the pattern, you can apply it to any function.

There are four main types of transformation you need to master for SL:

1. Translations (horizontal and vertical shifts)
2. Reflections (in the x-axis or y-axis)
3. Vertical stretches/compressions
4. Horizontal stretches/compressions

Transformations are described using the general form:

$y=pf(qx-a)+b$

This is equivalent to writing:

$y=pf\left(q\left(x-\frac{a}{q}\right)\right)+b$

The factored form makes it clear that the horizontal translation is $\frac{a}{q}$ units (not simply $a$ units) whenever $q\ne 1$. We will use this factored form throughout to avoid errors.

Vertical Translation: A transformation of the form $y=f(x)+b$ that shifts every point on the graph of $f(x)$ up or down by $|b|$ units, without changing its shape.

The rule is straightforward:

• If $b>0$, the graph shifts up by $b$ units.
• If $b<0$, the graph shifts down by $|b|$ units.

Every point $(x,y)$ on the original graph maps to $(x,y+b)$.