Combined Transformations

All the transformations you have learned so far can be combined into a single, powerful equation:

$$y = af(b(x - h)) + k$$

Each letter in this equation controls a specific transformation:

• $h$: horizontal translation ($h > 0$ means right, $h < 0$ means left)
• $b$: horizontal dilation by factor $\frac{1}{|b|}$ from the $y$-axis; if $b < 0$, there is also a reflection in the $y$-axis
• $a$: vertical dilation by factor $|a|$ from the $x$-axis; if $a < 0$, there is also a reflection in the $x$-axis
• $k$: vertical translation ($k > 0$ means up, $k < 0$ means down)

Reading transformations from the equation

The safest way to read combined transformations is from the inside out:

1. Start with the innermost change: $x - h$ tells you the horizontal translation
2. Next, $b(x - h)$ tells you the horizontal dilation (and any $y$-axis reflection)
3. Then, $af(\dots)$ tells you the vertical dilation (and any $x$-axis reflection)
4. Finally, $+ k$ tells you the vertical translation

Order matters (sometimes)

Some transformations can be applied in any order without changing the final result:

• Horizontal and vertical translations commute with each other
• Horizontal and vertical dilations commute with each other

However, translations and dilations along the same axis generally do not commute. That is why we write the equation in the standard form above — the parentheses fix the correct order.

General form: $y = af(b(x - h)) + k$; $a$ = vertical dilation + $x$-axis reflection if $a < 0$; $b$ = horizontal dilation + $y$-axis reflection if $b < 0$

Pause — copy the general transformation form $y = af(b(x-h)) + k$ with the role of each parameter ($a$: vertical dilation/reflection; $b$: horizontal dilation/reflection; $h$: horizontal shift; $k$: vertical shift) into your book.

We just saw that $y = af(b(x-h)) + k$ encodes all four transformations in one compact expression. That raises a question: if I know a point $(x_0, y_0)$ on the original graph, where does it land on the transformed graph? This card answers it → the mapping formula $x_{\text{new}} = \frac{x_0}{b} + h$, $y_{\text{new}} = ay_0 + k$.

If you know a point $(x, y)$ on the original graph $y = f(x)$, you can find its image on the transformed graph $y = af(b(x - h)) + k$ using the following formula.

If $(x, y)$ is on $y = f(x)$, the corresponding point on the transformed graph is found by:

• The input to the transformed function that produces the same inner value is $x_{\text{new}}$ where $b(x_{\text{new}} - h) = x$, so $x_{\text{new}} = \frac{x}{b} + h$
• The output is $y_{\text{new}} = ay + k$

So the correct transformed point is:

$$\left(\frac{x}{b} + h,\; ay + k\right)$$

To map a point $(x, y)$ on $y = f(x)$ to the transformed graph $y = af(b(x-h))+k$:; $x_{\text{new}} = \dfrac{x}{b} + h$ (divide by $b$ first, then add $h$)

Pause — copy the point-mapping formulas: $x_{\text{new}} = \dfrac{x}{b} + h$ and $y_{\text{new}} = ay + k$, and the memory cue (divide $x$ by $b$ first, then add $h$) into your book.