Transformation Rules and Composite Transformations

Transformation rules

In the basic table, $a>0$. For the composite rule $y=a f(bx+c)+d$, require $a\ne0$ and $b\ne0$: use scale factors $|a|$ and $1/|b|$, adding the appropriate reflection when a multiplier is negative. The signed translations are horizontal displacement $-c$ before scaling and vertical displacement $d$.

\(y=\mathrm{f}(x)\)	Horizontal Transformation	Vertical Transformation
Translation	\(a\) units in the positive \(x\) direction: Replace \(x\) with \(x-a\) \(y=\mathrm{f}(x-a)\)	\(a\) units in the positive \(y\) direction: Replace \(y\) with \(y-a\) \(y-a=\mathrm{f}(x) \Leftrightarrow y=\mathrm{f}(x)+a\)
\(a\) units in the negative \(x\) direction: Replace \(x\) with \(x+a\) \(y=\mathrm{f}(x+a)\)	\(a\) units in the negative \(y\) direction: Replace \(y\) with \(y+a\) \(y+a=\mathrm{f}(x) \Leftrightarrow y=\mathrm{f}(x)-a\)
Scaling	Scaling by a factor of \(a\) parallel to the \(x\)-axis: Replace \(x\) with \(\frac{x}{a}\) \(y=\mathrm{f}\left(\frac{x}{a}\right)\)	Scaling by a factor of \(a\) parallel to the \(y\)-axis: Replace \(y\) with \(\frac{y}{a}\) \(\frac{y}{a}=\mathrm{f}(x) \Leftrightarrow y=a\mathrm{f}(x)\)
Reflection	Reflection about the \(y\)-axis: Replace \(x\) with \(-x\) \(y=\mathrm{f}(-x)\)	Reflection about the \(x\)-axis: Replace \(y\) with \(-y\) \(-y=\mathrm{f}(x) \Leftrightarrow y=-\mathrm{f}(x)\)
Modulus	\(y=\mathrm{f}(\|x\|)\) Omit the negative \(x\) portion. (where \(x < 0\)) Retain the non-negative \(x\) portion of the graph and reflect it about the \(y\) axis.	\(y=\|\mathrm{f}(x)\|\) Reflect the negative \(y\) portion to positive about the \(x\) axis. Omit the negative \(y\) portion (where \(y < 0\)).

$y=\mathrm{f}(x)$

Horizontal Transformation

Vertical Transformation

Translation

$a$ units in the positive $x$ direction:

Replace $x$ with $x-a$

$y=\mathrm{f}(x-a)$

$a$ units in the positive $y$ direction:

Replace $y$ with $y-a$

$y-a=\mathrm{f}(x) \Leftrightarrow y=\mathrm{f}(x)+a$

$a$ units in the negative $x$ direction:

Replace $x$ with $x+a$

$y=\mathrm{f}(x+a)$

$a$ units in the negative $y$ direction:

Replace $y$ with $y+a$

$y+a=\mathrm{f}(x) \Leftrightarrow y=\mathrm{f}(x)-a$

Scaling

Scaling by a factor of $a$ parallel to the $x$-axis:

Replace $x$ with $\frac{x}{a}$

$y=\mathrm{f}\left(\frac{x}{a}\right)$

Scaling by a factor of $a$ parallel to the $y$-axis:

Replace $y$ with $\frac{y}{a}$

$\frac{y}{a}=\mathrm{f}(x) \Leftrightarrow y=a\mathrm{f}(x)$

Reflection

Reflection about the $y$-axis:

Replace $x$ with $-x$

$y=\mathrm{f}(-x)$

Reflection about the $x$-axis:

Replace $y$ with $-y$

$-y=\mathrm{f}(x) \Leftrightarrow y=-\mathrm{f}(x)$

Modulus

$y=\mathrm{f}(|x|)$

Omit the negative $x$ portion. (where $x < 0$)

Retain the non-negative $x$ portion of the graph and reflect it about the $y$ axis.

$y=|\mathrm{f}(x)|$

Reflect the negative $y$ portion to positive about the $x$ axis.

Omit the negative $y$ portion (where $y < 0$).

Composite transformations

Replacement method for $y=a\mathrm{f}(bx+c)+d$

\[y=\mathrm{f}(x)\xrightarrow{\mathrm{I}}y=\mathrm{f}(x+c)\xrightarrow{\mathrm{II}}y=\mathrm{f}(bx+c)\xrightarrow{\mathrm{III}}y=a\mathrm{f}(bx+c)\xrightarrow{\mathrm{IV}}y=a\mathrm{f}(bx+c)+d\]

Step	Transformation	Replacement
$\mathrm{I}$	Translate by signed horizontal displacement $-c$.	Replace $x$ with $x+c$.
$\mathrm{II}$	Scale by factor $1/\|b\|$ parallel to the $x$-axis; also reflect in the $y$-axis if $b<0$.	Replace $x$ with $bx$.
$\mathrm{III}$	Scale by factor $\|a\|$ parallel to the $y$-axis; also reflect in the $x$-axis if $a<0$.	Replace $y$ with $y/a$.
$\mathrm{IV}$	Translate by signed vertical displacement $d$.	Replace $y$ with $y-d$.

Replacement method for $y=\mathrm{f}(a-\lvert x\rvert)$

\[y=\mathrm{f}(x)\xrightarrow{\mathrm{I}}y=\mathrm{f}(x+a)\xrightarrow{\mathrm{II}}y=\mathrm{f}(-x+a)\xrightarrow{\mathrm{III}}y=\mathrm{f}(-\lvert x\rvert+a)\]

Step	Transformation	Replacement
$\mathrm{I}$	Translate $a$ units in the negative $x$-direction.	Replace $x$ with $x+a$.
$\mathrm{II}$	Reflect in the $y$-axis.	Replace $x$ with $-x$.
$\mathrm{III}$	Apply the horizontal modulus transformation.	Replace $x$ with $\lvert x\rvert$.

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Step	Transformation	Replacement
\(\mathrm{I}\)	Translate by signed horizontal displacement \(-c\).	Replace \(x\) with \(x+c\).
\(\mathrm{II}\)	Scale by factor \(1/\|b\|\) parallel to the \(x\)-axis; also reflect in the \(y\)-axis if \(b<0\).	Replace \(x\) with \(bx\).
\(\mathrm{III}\)	Scale by factor \(\|a\|\) parallel to the \(y\)-axis; also reflect in the \(x\)-axis if \(a<0\).	Replace \(y\) with \(y/a\).
\(\mathrm{IV}\)	Translate by signed vertical displacement \(d\).	Replace \(y\) with \(y-d\).

Step	Transformation	Replacement
\(\mathrm{I}\)	Translate \(a\) units in the negative \(x\)-direction.	Replace \(x\) with \(x+a\).
\(\mathrm{II}\)	Reflect in the \(y\)-axis.	Replace \(x\) with \(-x\).
\(\mathrm{III}\)	Apply the horizontal modulus transformation.	Replace \(x\) with \(\lvert x\rvert\).