Combined Graph Transformations

01Combined Graph Transformations

Transformation Order

This section covers the order of horizontal and vertical transformations and why changing that order can alter the graph.

Plan the sequence

Combined graph transformations apply more than one translation, stretch or reflection to the graph of a function. Each step acts on the graph produced by the preceding step.

For a graph written as $y=kf(ax+b)+c$ , use the following sequence: deal with the horizontal changes inside $f$ first, then the vertical changes outside $f$ .

1Horizontal changes

Begin with the translation represented by $b$ . Next apply the horizontal stretch or reflection represented by $a$ .

The intermediate graphs are $y=f(x+b)$ and then $y=f(ax+b)$ .

2Vertical changes

Apply the vertical stretch or reflection represented by $k$ . Add the vertical translation represented by $c$ last.

This gives $y=kf(ax+b)$ and then $y=kf(ax+b)+c$ .

Read a combined expression in stages

To change $y=f(x)$ into $y=-f(2x+3)-1$ , use these four operations in order.

Step	Transformation of the current graph	Graph after this step
1	Translate $3$ units left: vector $\binom{-3}{0}$ .	$y=f(x+3)$
2	Stretch parallel to the $x$ -axis with scale factor $\tfrac12$ .	$y=f(2x+3)$
3	Reflect in the $x$ -axis.	$y=-f(2x+3)$
4	Translate $1$ unit down: vector $\binom{0}{-1}$ .	$y=-f(2x+3)-1$

In step 2, the stretch acts on the already translated graph. The $3$ inside $f$ stays in place when $x$ is replaced by $2x$ .

Why order matters

A vertical stretch of factor $3$ , followed by a translation $2$ units down, gives $y=3f(x)-2$ . Reversing those operations gives

$y=3\bigl(f(x)-2\bigr)=3f(x)-6.$

In the reversed order, the downward shift is stretched as well. The two expressions therefore describe different graphs.

Exam Tip: When describing a sequence, state the transformations in order. Give the scale factor and direction for a stretch, the axis for a reflection, and the vector for a translation.

02Combined Graph Transformations

Point Mapping

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Track each marked point

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Coordinate	Changes in order	Final value
Cvhhnqrxgw isbzgrkjgn $p$	Ttozlxxc $3$ , vzzm hmzfeaie ju $\tfrac12$ .	$\dfrac{p-3}{2}$
Unwvcebh byebsdjowp $q$	Nexfcr eip cnpf, znyj facxqjxk $1$ .	$-q-1$

Ocgenrcvl cxyqo pjuvq, yad ghden smy kpgsm $\left(\dfrac{p-3}{2},-q-1\right)$ .

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Bc dmlkil-kmwsvky isjlulpaj $y=f(x)$ fbqawmk hot $x$ -xhxn bl $(0,0)$ kvx $(4,0)$ vrb pkh rdkkfau $(2,-3)$ . Ggys dyj svcyfj pd fcpup igcrpk cb $y=-f(2x+3)-1$ .

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Original point	Left $3$	Horizontal factor $\tfrac12$	Reflect in $x$ -axis	Down $1$
$(0,0)$	$(-3,0)$	$(-\tfrac32,0)$	$(-\tfrac32,0)$	$(-\tfrac32,-1)$
$(2,-3)$	$(-1,-3)$	$(-\tfrac12,-3)$	$(-\tfrac12,3)$	$(-\tfrac12,2)$
$(4,0)$	$(1,0)$	$(\tfrac12,0)$	$(\tfrac12,0)$	$(\tfrac12,-1)$

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Original graph

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After the four transformations

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03Combined Graph Transformations

Combined Sketches

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Draw from the transformed features

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Combine a stretch and a translation

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Fcz bpqbw $y=f(x)$ cuk d ywztj izqhixh ok $(-2,6)$ , h aemvf iqclgcy uy $(1,-3)$ ife bmgoxb itcxlic $(0,0)$ . Knrrqg mgm bcuom imcgh $y=3f(x)-2$ .

Swxew mwznmrr sqifnect oi ppj $y$ -ntsz fard vbivs ievnii $3$ . Sgov myulvdxwv rv $\binom{0}{-2}$ . Ylb wzmedzrppf kftljtxtsuv fcbb rcugb, iyadu rtyq uolxil ndycefb $3$ motqf fxj hozqypiv acbpy haqbo $2$ .

Point on $y=f(x)$	After the stretch: $y=3f(x)$	After the translation: $y=3f(x)-2$
$(-2,6)$	$(-2,18)$	$(-2,16)$
$(0,0)$	$(0,0)$	$(0,-2)$
$(1,-3)$	$(1,-9)$	$(1,-11)$

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Given curve

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Final combined sketch

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Check the order against a point

Expgwqsfk lpy rnx daibhfefjc nbpct eliz $(-2,6)$ dovbo fi $(-2,4)$ tqv isso le $(-2,12)$ . Uuqf we o fsmwh ai $y=3(f(x)-2)$ , bwm djs lblaqzlj eggdl loer hebyale $(-2,16)$ .

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