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The Order of Transformations

Example 1

The graph of $f\left(x\right)$ (shown) is transformed in the following order:
  1. A reflection in the $x$-axis
  2. A vertical stretch with scale factor $2$
  3. A translation by vector $\binom{2}{0}$
to produce the graph of $g\left(x\right)$.

In the Desmos window above, type $g\left(x\right)$ in terms of $f\left(x\right).\qquad g\left(x\right)=$

Example 2

The graph of $f\left(x\right)$ (black) is transformed to produce the graph of $g\left(x\right)$ (green).

In the Desmos window above, type $g\left(x\right)$ in terms of $f\left(x\right).\qquad g\left(x\right)=$

Example 3

The graph of $y=\sqrt{x}$ (shown) is transformed in the following order:
  1. A vertical compression with scale factor $\frac{1}{2}$
  2. A reflection in the $x$-axis
  3. A translation to the left by 4 units
to produce the graph of $g\left(x\right)$.

In the Desmos window above, type $g\left(x\right)$ in terms of $x.\qquad g\left(x\right)=$

Example 4

The graph of $f\left(x\right)$ (black) is transformed to produce the graph of $g\left(x\right)$ (green).

In the Desmos window above, type $g\left(x\right)$ in terms of $x.\qquad g\left(x\right)=$
*You do not need to simplify