Transformations

For non-linear, HL transformations, see composition of function and absolute value, and reciprocal and square of functions.

While functions are like families of equations, transformations of functions are like families of functions.

Transformations allow us to examine related functions without doing too much work. In particular, we shall apply linear transformations to $x$ and $y$, such that functions are stretched, reflected, or translated along the horizontal and vertical directions.

Contents

• Everything will make sense (I promise)
• Examples

Everything will make sense (I promise)

This derivation is very good to know, but it is not asked on exams.

Say we are familiar with some function $y_1 = f_1(x_1)$, and we want to learn more about some related $y_2 = f_2(x_2)$. The functions are related such that

Our goal is to rewrite $f_2$ using $f_1$, and we want to evaluate $f_1$ using some expression of $x_1$. An analogy is that if we are switching the locks, then we also have to switch the keys.

Then using $y_1 = f_1(x_1)$, we obtain

We shall refer $\lvert b\rvert$ and $\lvert a\rvert$ as the horizontal and vertical stretch factors. If $b$ and $a$ are negative, then there is also a reflection across the $y$- or $x$-axis. Finally $h$ and $k$ are horizontal and vertical translations, which are applied after stretches and/or reflections.

In other words, when expressing $f_2$ as a transformation of $f_1$, we apply the $x$-transformations in reverse, evaluate through $f_1$, then apply the $y$-transformations in the forward order.

HL candidates should be able to rewrite $f(3x + 5)$ as $f\left(3\left(x + \frac{5}{3}\right)\right)$ to identify the horizontal stretch factor $\frac13$ and horizontal shift of $\frac53$ units to the left.

Examples