SAT Function Transformations: Left, Right, Up, Down Without Memorizing Tricks

Learn SAT function transformations by reasoning from inputs and outputs: horizontal and vertical shifts, reflections, stretches, compressions, and common sign traps.

Reviewed by Aakash S. · Cornell University '14 · Counselling Advisor, World ViralPublished 29 September 2026Last updated 29 September 2026

Function-transformation rules are easy to mix up when they are memorized as a list.

The most common confusion is this:

f(x)+3

moves a graph up 3,

but

f(x+3)

moves it left 3.

Why does the inside sign appear to work “backward”?

Because outside changes affect the output, while inside changes affect the input required to reach the same output.

Once you understand that distinction, transformation questions become much more reliable.

Current SAT connection: function transformations

College Board's Advanced Math domain includes nonlinear functions and connections among equations, graphs, variables, factors, and transformations. On the SAT, a transformation may appear through an abstract function (f), a quadratic in vertex form, an exponential function, an absolute-value graph, or a table of input-output values.

Use SAT Functions for the parent topic, Vertex Form vs Standard Form for quadratic transformations, and SAT Domain and Range when a shift changes valid inputs or outputs.

The core idea: input vs output

A function takes an input $x$ and produces an output $f(x)$ .

There are two fundamentally different ways to alter it.

Change outside the function

f(x)+k

changes the output.

Every y-value becomes $k$ larger.

So the graph moves vertically.

Change inside the function

f(x-h)

changes which input produces a given output.

That moves the graph horizontally.

This is the source of the “opposite sign” behavior.

Transformation map

Starting from:

y=f(x),

the most common transformations are:

New function	Transformation
$f(x)+k$	up $k$
$f(x)-k$	down $k$
$f(x-h)$	right $h$
$f(x+h)$	left $h$
$-f(x)$	reflect across x-axis
$f(-x)$	reflect across y-axis
$af(x)$	vertical scale by $\lvert a\rvert$ ; reflect across x-axis if $a<0$
$f(bx)$	horizontal scale by factor $1/\lvert b\rvert$ ; reflect across y-axis if $b<0$

For many SAT questions, shifts and reflections are more important than the full scaling vocabulary.

Vertical shifts: easiest case

Suppose:

g(x)=f(x)+4.

For every input $x$ , the output is 4 larger than before.

If:

f(2)=7,

then:

g(2)=f(2)+4=11.

So every point:

(x,y)

on the original graph becomes:

(x,y+4).

The graph moves up 4.

Similarly:

f(x)-6

moves the graph down 6.

Worked example 1: vertical shift from a point

The point $(3,-2)$ lies on:

y=f(x).

What point lies on:

y=f(x)+5?

Same input:

x=3.

Output increases by 5:

-2+5=3.

So the new point is:

\boxed{(3,3)}.

Horizontal shifts: understand the input

Suppose:

g(x)=f(x-4).

Ask:

What input makes $g$ use the old input 2?

We need:

x-4=2.

So:

x=6.

That means whatever happened at $x=2$ in $f$ now happens at $x=6$ in $g$ .

The graph moved right 4.

This is why:

f(x-4)

means right 4.

Worked example 2: horizontal shift from a point

The point:

(2,7)

lies on:

y=f(x).

Find the corresponding point on:

y=f(x-5).

The graph moves right 5.

So:

(2,7)\rightarrow(7,7).

Answer: $(7,7)$ .

The y-value stays the same because the transformation is horizontal.

Why f(x+3) moves left

Suppose an important point occurs when the original input is:

x=10.

For:

g(x)=f(x+3),

we want:

x+3=10.

Then:

x=7.

The same output now occurs 3 units earlier on the x-axis.

So the graph moves left 3.

No slogan required.

Point-mapping method

If:

(x,y)

is on:

y=f(x),

then:

f(x)+k

(x,y)\rightarrow(x,y+k).

f(x)−k

(x,y)\rightarrow(x,y-k).

f(x−h)

(x,y)\rightarrow(x+h,y).

f(x+h)

(x,y)\rightarrow(x-h,y).

This method is extremely useful when an SAT question provides one or two points instead of a complete graph.

Worked example 3: combine horizontal and vertical shifts

Suppose:

(1,4)

lies on:

y=f(x).

What corresponding point lies on:

y=f(x-3)+6?

First:

f(x-3)

moves right 3:

(1,4)\rightarrow(4,4).

Then:

+6

moves up 6:

(4,4)\rightarrow(4,10).

Answer: $(4,10)$ .

Transformations of a quadratic

Start with:

f(x)=x^2.

Then:

g(x)=(x-4)^2+7.

Compared with $y=x^2$ :

right 4;
up 7.

The vertex moves from:

(0,0)

to:

(4,7).

This is exactly why vertex form:

a(x-h)^2+k

has vertex:

(h,k).

Function transformations and quadratic vertex form are the same idea viewed from two directions.

Worked example 4: read a parabola transformation

Describe:

y=-2(x+3)^2+5

relative to:

y=x^2.

Start inside:

x+3

means left 3.

Outside:

+5

means up 5.

Coefficient:

-2

does two things:

negative reflects across the x-axis;
magnitude 2 vertically stretches the graph.

So the vertex is:

(-3,5),

and the parabola opens downward.

Reflections across the x-axis

If:

g(x)=-f(x),

every y-value changes sign.

So:

(x,y)\rightarrow(x,-y).

That reflects the graph across the x-axis.

Example:

If:

f(4)=9,

then:

-f(4)=-9.

Point:

(4,9)

becomes:

(4,-9).

Reflections across the y-axis

If:

g(x)=f(-x),

the input sign changes.

Point:

(x,y)

becomes:

(-x,y).

That reflects the graph across the y-axis.

Example:

If $(5,2)$ lies on $f$ , then $(-5,2)$ lies on $f(-x)$ .

Worked example 5: distinguish the two reflections

Suppose:

(3,-8)

lies on $y=f(x)$ .

On y=−f(x)

Reflect across x-axis:

(3,8).

On y=f(−x)

Reflect across y-axis:

(-3,-8).

The position of the negative sign tells you whether input or output changes.

Vertical scaling

If:

g(x)=3f(x),

every output is multiplied by 3.

Point:

(x,y)

becomes:

(x,3y).

Example:

(2,4)\rightarrow(2,12).

This is called a vertical stretch by factor 3.

If:

g(x)=\frac12f(x),

then:

(x,y)\rightarrow\left(x,\frac12y\right).

That is a vertical compression.

Negative vertical scaling

If:

g(x)=-3f(x),

then:

multiply y-values by 3;
reverse their signs.

So it is a vertical stretch plus reflection across the x-axis.

Example:

(2,4)\rightarrow(2,-12).

Horizontal scaling: the reciprocal effect

Suppose:

g(x)=f(2x).

If the old graph reaches some feature at:

x=6,

then the new graph reaches the same feature when:

2x=6.

So:

x=3.

The graph is horizontally compressed by a factor of:

\frac12.

That reciprocal relationship is why horizontal transformations often feel reversed.

Worked example 6: horizontal scale from a point

The point $(8,5)$ lies on $y=f(x)$ .

Find the corresponding point on:

y=f(2x).

We need:

2x=8.

So:

x=4.

Corresponding point:

\boxed{(4,5)}.

Horizontal stretch

For:

g(x)=f\left(\frac{x}{3}\right),

to reproduce the old input 4:

\frac{x}{3}=4.

So:

x=12.

Features move three times farther from the y-axis.

This is a horizontal stretch by factor 3.

Do you need to memorize horizontal scaling for the SAT?

You should understand it, but do not let advanced vocabulary distract you from the question.

Many transformation questions can be solved by:

substituting a known input;
mapping one point;
locating a vertex;
matching an equation to a graph.

If the reciprocal scaling rule feels fragile, derive it from:

What new x-value makes the inside equal the old x-value?

That always works.

Function notation transformation questions

Suppose:

f(3)=10.

What is:

g(7)

if:

g(x)=f(x-4)+2?

Substitute:

g(7)=f(7-4)+2

=f(3)+2.

Given:

f(3)=10,

we get:

g(7)=12.

This is the algebraic version of “right 4, up 2.”

Worked example 7: use a table

Suppose the table contains:

$x$	$f(x)$
-2	5
1	8
4	-3

Define:

g(x)=f(x+3)-2.

Find:

g(1).

g(1)=f(4)-2.

From the table:

f(4)=-3.

So:

g(1)=-5.

No graph is needed.

Transformations and domain

Transformations can also shift the domain.

Suppose:

f(x)=\sqrt{x}.

Its domain is:

x\ge0.

Now:

g(x)=\sqrt{x-5}.

Require:

x-5\ge0.

So:

x\ge5.

This matches the graph shift: the square-root graph moved right 5.

Transformations and absolute value

Start with:

f(x)=|x|.

Then:

g(x)=|x+2|-7.

That moves:

left 2;
down 7.

Vertex moves:

(0,0)\rightarrow(-2,-7).

Again, the same transformation rules work across different function families.

Transformations and exponential functions

Start:

f(x)=2^x.

Then:

g(x)=2^{x-3}+4.

That moves the exponential graph:

right 3;
up 4.

The horizontal asymptote moves from:

y=0

to:

y=4.

The rules are not specific to quadratics.

A deeper distinction: changing x vs changing f(x)

Consider:

f(x-2)+5.

There are two layers.

Inside:

x-2

changes which original input is being used.

Outside:

+5

changes what output is produced afterward.

This order-of-operations view is more powerful than memorizing arrows.

Common trap 1: reading inside signs normally

Incorrect:

f(x-6)\Rightarrow\text{left 6}.

Correct:

f(x-6)\Rightarrow\text{right 6}.

Reason:

To reproduce old input $x_0$ , the new input must satisfy:

x-6=x_0

so:

x=x_0+6.

Common trap 2: applying an outside number to x-values

For:

f(x)+4,

x-values do not change.

Only y-values do.

Common trap 3: confusing −f(x) and f(−x)

-f(x)

reflects vertically across the x-axis.

f(-x)

reflects horizontally across the y-axis.

Think:

minus outside → output sign changes;
minus inside → input sign changes.

Common trap 4: thinking 2f(x) shifts up 2

It does not.

It multiplies every y-value by 2.

That is scaling, not translation.

Compare:

2f(x)

with:

f(x)+2.

Very different graphs.

Common trap 5: thinking f(2x) stretches by 2

It actually compresses horizontally by factor:

\frac12.

Derive rather than memorize:

Old feature at $x=8$ .

Need:

2x=8

so new feature is at:

x=4.

Closer to y-axis.

Common trap 6: losing transformations in a rewritten equation

Suppose:

y=(x-3)^2+5.

If you expand:

y=x^2-6x+14,

the same graph remains, but the vertex is less obvious.

If the question is about transformations or the vertex, preserve vertex form.

Common trap 7: assuming every coefficient is a shift

In:

y=4(x-2)^2-1,

the 4 is not a horizontal or vertical shift.

It scales the output.

The shifts are:

right 2;
down 1.

Worked example 8: reconstruct a transformed function

The graph of $g$ is obtained from $f$ by shifting $f$ left 4 units and down 3 units.

Which equation represents $g$ ?

Left 4:

f(x+4).

Down 3:

f(x+4)-3.

So:

\boxed{g(x)=f(x+4)-3}.

Worked example 9: infer a shift from points

The point $(2,6)$ on $f$ corresponds to $(7,1)$ on $g$ .

Assume $g$ is only a translation of $f$ .

Horizontal change:

7-2=5.

So shift right 5.

Vertical change:

1-6=-5.

So shift down 5.

Therefore:

\boxed{g(x)=f(x-5)-5}.

Worked example 10: transformation and maximum

Suppose $f$ has a maximum value of 12.

Define:

g(x)=f(x-7)-4.

The horizontal shift does not change output values.

The vertical shift subtracts 4 from every output.

So the maximum of $g$ is:

12-4=8.

Answer: 8.

You do not need to know where the maximum occurs.

Worked example 11: transformation and root

Suppose:

f(3)=0.

Define:

g(x)=f(x+5).

For $g(x)=0$ , we need:

x+5=3.

So:

x=-2.

The root moved left 5.

Worked example 12: transformation and y-value

Suppose:

f(-1)=7.

If:

g(x)=-2f(x)+3,

then:

g(-1)=-2(7)+3=-11.

Again, outside operations act on outputs.

A reliable four-question method

When you see a transformed function, ask:

What changed inside the parentheses?
- affects x/input/horizontal position.
What changed outside the function?
- affects y/output/vertical position.
Is there a negative sign?
- outside → x-axis reflection;
- inside → y-axis reflection.
Is there multiplication?
- outside → vertical scale;
- inside → reciprocal horizontal scale.

This keeps transformations organized even in complicated expressions.

Mini drill

1.

g(x)=f(x-8)

Transformation: right 8.

2.

g(x)=f(x)+8

Transformation: up 8.

3.

g(x)=f(x+2)-5

Transformation: left 2, down 5.

4.

g(x)=-f(x)

Transformation: reflect across x-axis.

5.

g(x)=f(-x)

Transformation: reflect across y-axis.

6.

Point $(4,9)$ lies on $f$ . What point lies on $f(x-3)+2$ ?

Right 3, up 2:

\boxed{(7,11)}.

7.

If $f(5)=12$ and $g(x)=f(x+4)-7$ , find $g(1)$ .

g(1)=f(5)-7=5.

Answer: 5.

FAQ: SAT function transformations

Why does (f(x-3)) move right 3?

To reproduce the old input (a), the new input must satisfy (x-3=a), so (x=a+3). The feature appears 3 units to the right.

Why does (f(x)+3) move up 3?

The +3 changes every output directly, so every y-value rises by 3.

What is the difference between (-f(x)) and (f(-x))?

(-f(x)) reflects across the x-axis. (f(-x)) reflects across the y-axis.

Does (2f(x)) shift the graph up 2?

No. It multiplies all y-values by 2, creating a vertical stretch.

What does (f(2x)) do?

It horizontally compresses features toward the y-axis by a factor of (1/2).

Do transformation rules work for quadratics, exponentials, and absolute value?

Yes. The input/output logic is general. See SAT Quadratic Equations, SAT Exponential Growth and Decay, and SAT Absolute Value Equations.

Is Desmos useful for transformation questions?

Yes as a visual check, especially for complicated functions, but point-mapping or vertex recognition is often faster.

Final takeaway

Do not memorize transformations as an isolated sign chart.

Use the function machine:

outside changes the output;
inside changes the input needed to produce the same output.

That explains why:

f(x)+k

moves vertically in the expected direction, while:

f(x-h)

moves horizontally in the opposite-looking direction.

For SAT questions, that reasoning is more durable than a list of arrows—and it works for quadratics, absolute value, radicals, exponentials, tables, and abstract function notation.

How to practice this on MastaPrep

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SAT Function Transformations: Left, Right, Up, Down Without Memorizing Tricks

Learn SAT function transformations by reasoning from inputs and outputs: horizontal and vertical shifts, reflections, stretches, compressions, and common sign traps.

Reviewed by Aakash S. · Cornell University '14 · Counselling Advisor, World ViralPublished 29 September 2026Last updated 29 September 2026

Function-transformation rules are easy to mix up when they are memorized as a list.

The most common confusion is this:

f(x)+3

moves a graph up 3,

but

f(x+3)

moves it left 3.

Why does the inside sign appear to work “backward”?

Because outside changes affect the output, while inside changes affect the input required to reach the same output.

Once you understand that distinction, transformation questions become much more reliable.

Current SAT connection: function transformations

Use SAT Functions for the parent topic, Vertex Form vs Standard Form for quadratic transformations, and SAT Domain and Range when a shift changes valid inputs or outputs.

The core idea: input vs output

A function takes an input $x$ and produces an output $f(x)$ .

There are two fundamentally different ways to alter it.

Change outside the function

f(x)+k

changes the output.

Every y-value becomes $k$ larger.

So the graph moves vertically.

Change inside the function

f(x-h)

changes which input produces a given output.

That moves the graph horizontally.

This is the source of the “opposite sign” behavior.

Transformation map

Starting from:

y=f(x),

the most common transformations are:

New function	Transformation
$f(x)+k$	up $k$
$f(x)-k$	down $k$
$f(x-h)$	right $h$
$f(x+h)$	left $h$
$-f(x)$	reflect across x-axis
$f(-x)$	reflect across y-axis
$af(x)$	vertical scale by $\lvert a\rvert$ ; reflect across x-axis if $a<0$
$f(bx)$	horizontal scale by factor $1/\lvert b\rvert$ ; reflect across y-axis if $b<0$

For many SAT questions, shifts and reflections are more important than the full scaling vocabulary.

Vertical shifts: easiest case

Suppose:

g(x)=f(x)+4.

For every input $x$ , the output is 4 larger than before.

If:

f(2)=7,

then:

g(2)=f(2)+4=11.

So every point:

(x,y)

on the original graph becomes:

(x,y+4).

The graph moves up 4.

Similarly:

f(x)-6

moves the graph down 6.

Worked example 1: vertical shift from a point

The point $(3,-2)$ lies on:

y=f(x).

What point lies on:

y=f(x)+5?

Same input:

x=3.

Output increases by 5:

-2+5=3.

So the new point is:

\boxed{(3,3)}.

Horizontal shifts: understand the input

Suppose:

g(x)=f(x-4).

Ask:

What input makes $g$ use the old input 2?

We need:

x-4=2.

So:

x=6.

That means whatever happened at $x=2$ in $f$ now happens at $x=6$ in $g$ .

The graph moved right 4.

This is why:

f(x-4)

means right 4.

Worked example 2: horizontal shift from a point

The point:

(2,7)

lies on:

y=f(x).

Find the corresponding point on:

y=f(x-5).

The graph moves right 5.

So:

(2,7)\rightarrow(7,7).

Answer: $(7,7)$ .

The y-value stays the same because the transformation is horizontal.

Why f(x+3) moves left

Suppose an important point occurs when the original input is:

x=10.

For:

g(x)=f(x+3),

we want:

x+3=10.

Then:

x=7.

The same output now occurs 3 units earlier on the x-axis.

So the graph moves left 3.

No slogan required.

Point-mapping method

If:

(x,y)

is on:

y=f(x),

then:

f(x)+k

(x,y)\rightarrow(x,y+k).

f(x)−k

(x,y)\rightarrow(x,y-k).

f(x−h)

(x,y)\rightarrow(x+h,y).

f(x+h)

(x,y)\rightarrow(x-h,y).

This method is extremely useful when an SAT question provides one or two points instead of a complete graph.

Worked example 3: combine horizontal and vertical shifts

Suppose:

(1,4)

lies on:

y=f(x).

What corresponding point lies on:

y=f(x-3)+6?

First:

f(x-3)

moves right 3:

(1,4)\rightarrow(4,4).

Then:

+6

moves up 6:

(4,4)\rightarrow(4,10).

Answer: $(4,10)$ .

Transformations of a quadratic

Start with:

f(x)=x^2.

Then:

g(x)=(x-4)^2+7.

Compared with $y=x^2$ :

right 4;
up 7.

The vertex moves from:

(0,0)

to:

(4,7).

This is exactly why vertex form:

a(x-h)^2+k

has vertex:

(h,k).

Function transformations and quadratic vertex form are the same idea viewed from two directions.

Worked example 4: read a parabola transformation

Describe:

y=-2(x+3)^2+5

relative to:

y=x^2.

Start inside:

x+3

means left 3.

Outside:

+5

means up 5.

Coefficient:

-2

does two things:

negative reflects across the x-axis;
magnitude 2 vertically stretches the graph.

So the vertex is:

(-3,5),

and the parabola opens downward.

Reflections across the x-axis

If:

g(x)=-f(x),

every y-value changes sign.

So:

(x,y)\rightarrow(x,-y).

That reflects the graph across the x-axis.

Example:

If:

f(4)=9,

then:

-f(4)=-9.

Point:

(4,9)

becomes:

(4,-9).

Reflections across the y-axis

If:

g(x)=f(-x),

the input sign changes.

Point:

(x,y)

becomes:

(-x,y).

That reflects the graph across the y-axis.

Example:

If $(5,2)$ lies on $f$ , then $(-5,2)$ lies on $f(-x)$ .

Worked example 5: distinguish the two reflections

Suppose:

(3,-8)

lies on $y=f(x)$ .

On y=−f(x)

Reflect across x-axis:

(3,8).

On y=f(−x)

Reflect across y-axis:

(-3,-8).

The position of the negative sign tells you whether input or output changes.

Vertical scaling

If:

g(x)=3f(x),

every output is multiplied by 3.

Point:

(x,y)

becomes:

(x,3y).

Example:

(2,4)\rightarrow(2,12).

This is called a vertical stretch by factor 3.

If:

g(x)=\frac12f(x),

then:

(x,y)\rightarrow\left(x,\frac12y\right).

That is a vertical compression.

Negative vertical scaling

If:

g(x)=-3f(x),

then:

multiply y-values by 3;
reverse their signs.

So it is a vertical stretch plus reflection across the x-axis.

Example:

(2,4)\rightarrow(2,-12).

Horizontal scaling: the reciprocal effect

Suppose:

g(x)=f(2x).

If the old graph reaches some feature at:

x=6,

then the new graph reaches the same feature when:

2x=6.

So:

x=3.

The graph is horizontally compressed by a factor of:

\frac12.

That reciprocal relationship is why horizontal transformations often feel reversed.

Worked example 6: horizontal scale from a point

The point $(8,5)$ lies on $y=f(x)$ .

Find the corresponding point on:

y=f(2x).

We need:

2x=8.

So:

x=4.

Corresponding point:

\boxed{(4,5)}.

Horizontal stretch

For:

g(x)=f\left(\frac{x}{3}\right),

to reproduce the old input 4:

\frac{x}{3}=4.

So:

x=12.

Features move three times farther from the y-axis.

This is a horizontal stretch by factor 3.

Do you need to memorize horizontal scaling for the SAT?

You should understand it, but do not let advanced vocabulary distract you from the question.

Many transformation questions can be solved by:

substituting a known input;
mapping one point;
locating a vertex;
matching an equation to a graph.

If the reciprocal scaling rule feels fragile, derive it from:

What new x-value makes the inside equal the old x-value?

That always works.

Function notation transformation questions

Suppose:

f(3)=10.

What is:

g(7)

if:

g(x)=f(x-4)+2?

Substitute:

g(7)=f(7-4)+2

=f(3)+2.

Given:

f(3)=10,

we get:

g(7)=12.

This is the algebraic version of “right 4, up 2.”

Worked example 7: use a table

Suppose the table contains:

$x$	$f(x)$
-2	5
1	8
4	-3

Define:

g(x)=f(x+3)-2.

Find:

g(1).

g(1)=f(4)-2.

From the table:

f(4)=-3.

So:

g(1)=-5.

No graph is needed.

Transformations and domain

Transformations can also shift the domain.

Suppose:

f(x)=\sqrt{x}.

Its domain is:

x\ge0.

Now:

g(x)=\sqrt{x-5}.

Require:

x-5\ge0.

So:

x\ge5.

This matches the graph shift: the square-root graph moved right 5.

Transformations and absolute value

Start with:

f(x)=|x|.

Then:

g(x)=|x+2|-7.

That moves:

left 2;
down 7.

Vertex moves:

(0,0)\rightarrow(-2,-7).

Again, the same transformation rules work across different function families.

Transformations and exponential functions

Start:

f(x)=2^x.

Then:

g(x)=2^{x-3}+4.

That moves the exponential graph:

right 3;
up 4.

The horizontal asymptote moves from:

y=0

to:

y=4.

The rules are not specific to quadratics.

A deeper distinction: changing x vs changing f(x)

Consider:

f(x-2)+5.

There are two layers.

Inside:

x-2

changes which original input is being used.

Outside:

+5

changes what output is produced afterward.

This order-of-operations view is more powerful than memorizing arrows.

Common trap 1: reading inside signs normally

Incorrect:

f(x-6)\Rightarrow\text{left 6}.

Correct:

f(x-6)\Rightarrow\text{right 6}.

Reason:

To reproduce old input $x_0$ , the new input must satisfy:

x-6=x_0

so:

x=x_0+6.

Common trap 2: applying an outside number to x-values

For:

f(x)+4,

x-values do not change.

Only y-values do.

Common trap 3: confusing −f(x) and f(−x)

-f(x)

reflects vertically across the x-axis.

f(-x)

reflects horizontally across the y-axis.

Think:

minus outside → output sign changes;
minus inside → input sign changes.

Common trap 4: thinking 2f(x) shifts up 2

It does not.

It multiplies every y-value by 2.

That is scaling, not translation.

Compare:

2f(x)

with:

f(x)+2.

Very different graphs.

Common trap 5: thinking f(2x) stretches by 2

It actually compresses horizontally by factor:

\frac12.

Derive rather than memorize:

Old feature at $x=8$ .

Need:

2x=8

so new feature is at:

x=4.

Closer to y-axis.

Common trap 6: losing transformations in a rewritten equation

Suppose:

y=(x-3)^2+5.

If you expand:

y=x^2-6x+14,

the same graph remains, but the vertex is less obvious.

If the question is about transformations or the vertex, preserve vertex form.

Common trap 7: assuming every coefficient is a shift

In:

y=4(x-2)^2-1,

the 4 is not a horizontal or vertical shift.

It scales the output.

The shifts are:

right 2;
down 1.

Worked example 8: reconstruct a transformed function

The graph of $g$ is obtained from $f$ by shifting $f$ left 4 units and down 3 units.

Which equation represents $g$ ?

Left 4:

f(x+4).

Down 3:

f(x+4)-3.

So:

\boxed{g(x)=f(x+4)-3}.

Worked example 9: infer a shift from points

The point $(2,6)$ on $f$ corresponds to $(7,1)$ on $g$ .

Assume $g$ is only a translation of $f$ .

Horizontal change:

7-2=5.

So shift right 5.

Vertical change:

1-6=-5.

So shift down 5.

Therefore:

\boxed{g(x)=f(x-5)-5}.

Worked example 10: transformation and maximum

Suppose $f$ has a maximum value of 12.

Define:

g(x)=f(x-7)-4.

The horizontal shift does not change output values.

The vertical shift subtracts 4 from every output.

So the maximum of $g$ is:

12-4=8.

Answer: 8.

You do not need to know where the maximum occurs.

Worked example 11: transformation and root

Suppose:

f(3)=0.

Define:

g(x)=f(x+5).

For $g(x)=0$ , we need:

x+5=3.

So:

x=-2.

The root moved left 5.

Worked example 12: transformation and y-value

Suppose:

f(-1)=7.

If:

g(x)=-2f(x)+3,

then:

g(-1)=-2(7)+3=-11.

Again, outside operations act on outputs.

A reliable four-question method

When you see a transformed function, ask:

What changed inside the parentheses?
- affects x/input/horizontal position.
What changed outside the function?
- affects y/output/vertical position.
Is there a negative sign?
- outside → x-axis reflection;
- inside → y-axis reflection.
Is there multiplication?
- outside → vertical scale;
- inside → reciprocal horizontal scale.

This keeps transformations organized even in complicated expressions.

Mini drill

1.

g(x)=f(x-8)

Transformation: right 8.

2.

g(x)=f(x)+8

Transformation: up 8.

3.

g(x)=f(x+2)-5

Transformation: left 2, down 5.

4.

g(x)=-f(x)

Transformation: reflect across x-axis.

5.

g(x)=f(-x)

Transformation: reflect across y-axis.

6.

Point $(4,9)$ lies on $f$ . What point lies on $f(x-3)+2$ ?

Right 3, up 2:

\boxed{(7,11)}.

7.

If $f(5)=12$ and $g(x)=f(x+4)-7$ , find $g(1)$ .

g(1)=f(5)-7=5.

Answer: 5.

FAQ: SAT function transformations

Why does (f(x-3)) move right 3?

To reproduce the old input (a), the new input must satisfy (x-3=a), so (x=a+3). The feature appears 3 units to the right.

Why does (f(x)+3) move up 3?

The +3 changes every output directly, so every y-value rises by 3.

What is the difference between (-f(x)) and (f(-x))?

(-f(x)) reflects across the x-axis. (f(-x)) reflects across the y-axis.

Does (2f(x)) shift the graph up 2?

No. It multiplies all y-values by 2, creating a vertical stretch.

What does (f(2x)) do?

It horizontally compresses features toward the y-axis by a factor of (1/2).

Do transformation rules work for quadratics, exponentials, and absolute value?

Yes. The input/output logic is general. See SAT Quadratic Equations, SAT Exponential Growth and Decay, and SAT Absolute Value Equations.

Is Desmos useful for transformation questions?

Yes as a visual check, especially for complicated functions, but point-mapping or vertex recognition is often faster.

Final takeaway

Do not memorize transformations as an isolated sign chart.

Use the function machine:

outside changes the output;
inside changes the input needed to produce the same output.

That explains why:

f(x)+k

moves vertically in the expected direction, while:

f(x-h)

moves horizontally in the opposite-looking direction.

For SAT questions, that reasoning is more durable than a list of arrows—and it works for quadratics, absolute value, radicals, exponentials, tables, and abstract function notation.

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Current SAT connection: function transformations

The core idea: input vs output

Change outside the function

Change inside the function

Transformation map

Vertical shifts: easiest case

Worked example 1: vertical shift from a point

Horizontal shifts: understand the input

Worked example 2: horizontal shift from a point

Why f(x+3) moves left

Point-mapping method

f(x)+k

f(x)−k

f(x−h)

f(x+h)

Worked example 3: combine horizontal and vertical shifts

Transformations of a quadratic

Worked example 4: read a parabola transformation

Reflections across the x-axis

Reflections across the y-axis

Worked example 5: distinguish the two reflections

On y=−f(x)

On y=f(−x)

Vertical scaling

Negative vertical scaling

Horizontal scaling: the reciprocal effect

Worked example 6: horizontal scale from a point

Horizontal stretch

Do you need to memorize horizontal scaling for the SAT?

Function notation transformation questions

Worked example 7: use a table

Transformations and domain

Transformations and absolute value

Transformations and exponential functions

A deeper distinction: changing x vs changing f(x)

Common trap 1: reading inside signs normally

Common trap 2: applying an outside number to x-values

Common trap 3: confusing −f(x) and f(−x)

Common trap 4: thinking 2f(x) shifts up 2

Common trap 5: thinking f(2x) stretches by 2

Common trap 6: losing transformations in a rewritten equation

Common trap 7: assuming every coefficient is a shift

Worked example 8: reconstruct a transformed function

Worked example 9: infer a shift from points

Worked example 10: transformation and maximum

Worked example 11: transformation and root

Worked example 12: transformation and y-value

A reliable four-question method

Mini drill

1.

2.

3.

4.

5.

6.

7.

FAQ: SAT function transformations

Why does (f(x-3)) move right 3?

Why does (f(x)+3) move up 3?

What is the difference between (-f(x)) and (f(-x))?

Does (2f(x)) shift the graph up 2?

What does (f(2x)) do?

Do transformation rules work for quadratics, exponentials, and absolute value?

Is Desmos useful for transformation questions?

Final takeaway

How to practice this on MastaPrep

Related MastaPrep guides

Read next

Current SAT connection: function transformations

The core idea: input vs output

Change outside the function

Change inside the function

Transformation map

Vertical shifts: easiest case

Worked example 1: vertical shift from a point

Horizontal shifts: understand the input

Worked example 2: horizontal shift from a point

Why f(x+3) moves left

Point-mapping method

f(x)+k