Stretch then shift is not shift then stretch

stretch, then shift

2at x = 0, order A

6at x = 0, order B

4 apartverdict

The same two transformations, in two orders. The gap is 4 everywhere, not just at the vertex.

The figure needs JavaScript. It draws y = x² transformed two ways: stretch by 3 then up 2 gives 3x² + 2, passing through (0, 2); up 2 then stretch by 3 gives 3x² + 6, passing through (0, 6). The gap is 4 at every x. Later views show y = f(x − 3) moving the graph 3 to the right and y = f(2x) squeezing it horizontally by a factor of a half. Everything below stands on its own.

Why the order matters

Write each order as a chain and the reason appears.

Stretch 3, then up 2. x² becomes 3x², then 3x² + 2. The 2 is added after the stretching, so it is not stretched. At x = 0: 2.
Up 2, then stretch 3. x² becomes x² + 2, then 3(x² + 2) = 3x² + 6. The 2 was there when the stretching happened, so it got multiplied by 3 too.

The two differ by 4, which is 3 × 2 minus 2, at every value of x. So this is not a rounding difference or a special case at the vertex: the two curves are parallel and four apart for ever.

A vertical stretch multiplies everything that is already there. That is the whole rule. If a question gives you a sequence of transformations, do them in the stated order and write each line down, because the intermediate expression is where the information lives.

The four transformations

Written as	Does what	Check it with
y = f(x) + b	Moves up by b. Down if b is negative.	Every output gains b, so the whole graph lifts.
y = f(x − a)	Moves right by a.	To get the old output you now need a bigger x, so the picture slides right.
y = −f(x)	Reflects in the x-axis.	Outputs change sign. The x-axis does not move.
y = f(−x)	Reflects in the y-axis.	Inputs change sign. The y-axis does not move.
y = p f(x)	Vertical stretch, scale factor p.	Distances from the x-axis multiply by p.
y = f(qx)	Horizontal stretch, scale factor 1/q.	f(2x) squeezes to half width, because x = 6 now does what x = 3 did.

y = f(x − 3) moves the graph to the RIGHT. Almost everyone expects left, because of the minus. Here is the one sentence that fixes it: the new graph at x = 3 does what the old one did at x = 0, so whatever was at 0 is now at 3. The inside of a function works against you, which is also why f(2x) squeezes instead of stretching.

Horizontal, and the factor people invert

y = f(2x) has scale factor ½, not 2.

The new graph at x = 3 computes f(6).
So the feature that was at x = 6 now appears at x = 3.
Everything is half as far out: the graph is squashed towards the y-axis.

And y = f(x/2) has scale factor 2, stretching it away. The number inside and the effect are reciprocals, every time.

Translation as a vector

A translation is often given as a column vector. The vector (3, −2), written vertically, means:

3 to the right, which inside the function is f(x − 3)
2 down, which outside the function is − 2

So the whole thing is y = f(x − 3) − 2. Note the sign flip on the horizontal component and not on the vertical one. The vector is the honest description; the formula is where the signs get confusing.

On the GDC: two orders on one screen

Graphing both orders together makes the four-unit gap a thing you can see, and it takes about twenty seconds.

When you may use it. Applications. A calculator is allowed in every paper, and a transformation question is one of the few places a quick plot settles an argument outright.

TI-Nspire CX II

ctrl doc → Add Graphs, then f1(x)=3x x² +2
tab and f2(x)=3(x x² +2)
Both curves appear, four apart. menu → Trace and walk along to confirm the gap does not change
For a sliding transformation: menu → Actions → Insert Slider, call it a, and graph f3(x)=(x-a) x² to watch it move right as a grows

Casio fx-CG50

MENU → Graph, with Y1=3X²+2 and Y2=3(X²+2)
F6 DRAW, then SHIFT F1 Trace and read both y values at the same x
For a sliding transformation: MENU → Dyna Graph, enter Y=(X-A)² and let A run from −3 to 3
Dyna Graph is the one mode worth knowing for this sub-topic: it animates the parameter rather than making you re-plot

The mark people lose. Writing the horizontal transformation with the wrong sign. A translation 3 to the right is f(x − 3), and students write f(x + 3) because the graph went the positive way. Test it on one point before committing: if the old graph had a vertex at x = 0 and the new one has it at x = 3, substitute x = 3 and see which version gives you the old output. The other one is giving a horizontal stretch factor of q instead of 1/q.

Your turn

1. Starting from y = x², stretch vertically by 3 and then translate up 2. What is y at x = 2?

2. Now translate up 2 first and then stretch by 3. What is y at x = 2?

3. A graph has a maximum at x = 0. After the transformation y = f(x − 3), where is the maximum?

4. A graph has a feature at x = 6. After y = f(2x), where is it?

5. Why do the two orders give different answers?

Question 5. Why do the two orders give different answers? Because the stretch is applied to the translation as well when the translation comes first. Because a vertical stretch and a vertical translation are both vertical, so they cannot be combined. They do not really differ; the difference is in how the graph is drawn. Because the second order reflects the graph as well.

Where the marks go

1 markEach transformation named correctly, with its direction or scale factor.

1 markThe intermediate expression, when there is more than one transformation.

1 markThe final equation, or the image of a named point.

A question that gives you two transformations in a stated order is testing the order. Write the middle line, even though you can see the end, because that line is what shows you did them in the order asked.