Transformations of Functions

Starting from the graph of a known function, we can obtain the graphs of many related functions by applying simple geometric operations: shifting, reflecting, and scaling. Mastering these transformations eliminates the need to re-plot every function from scratch.

Quick Reference

Transformation	Effect on graph	How each point $(a, b)$ moves
$y = f (x) + c$ , $c > 0$	Shift up $c$ units	$(a, b) \to (a, b + c)$
$y = f (x) - c$ , $c > 0$	Shift down $c$ units	$(a, b) \to (a, b - c)$
$y = f (x - c)$ , $c > 0$	Shift right $c$ units	$(a, b) \to (a + c, b)$
$y = f (x + c)$ , $c > 0$	Shift left $c$ units	$(a, b) \to (a - c, b)$
$y = - f (x)$	Reflect in $x$ -axis	$(a, b) \to (a, - b)$
$y = f (- x)$	Reflect in $y$ -axis	$(a, b) \to (- a, b)$
$y = c f (x)$ , $c > 1$	Vertical stretch by $c$	$(a, b) \to (a, c b)$
$y = c f (x)$ , $0 < c < 1$	Vertical compress by $c$	$(a, b) \to (a, c b)$
$y = f (c x)$ , $c > 1$	Horizontal compress by $1 / c$	$(a, b) \to (a / c, b)$
$y = f (c x)$ , $0 < c < 1$	Horizontal stretch by $1 / c$	$(a, b) \to (a / c, b)$
$y = \| f (x) \|$	Reflect negative part up	Points below $x$ -axis flip above
$y = f (\| x \|)$	Reflect right side across $y$ -axis	Left side replaced by mirror of right

Vertical Shifts: $y = f (x) \pm c$

To obtain the graph of $y = f (x) + c$ ( $c > 0$ ), shift the graph of $f$ upward by $c$ units.

To obtain the graph of $y = f (x) - c$ ( $c > 0$ ), shift the graph of $f$ downward by $c$ units.

Diagram showing vertical shifts of f(x): upward to f(x)+c and downward to f(x)-c. — Adding a positive constant $c$ shifts the graph up; subtracting shifts it down.

Horizontal Shifts: $y = f (x \mp c)$

To obtain the graph of $y = f (x - c)$ ( $c > 0$ ), shift the graph of $f$ right by $c$ units.

To obtain the graph of $y = f (x + c)$ ( $c > 0$ ), shift the graph of $f$ left by $c$ units.

Key reminder: subtracting from the input shifts the graph right; adding to the input shifts it left. This is the opposite of what you might expect.

Diagram showing horizontal shifts of f(x): right to f(x-c) and left to f(x+c). — Horizontal shifts are counterintuitive: $f (x - c)$ shifts right, $f (x + c)$ shifts left.

Use the graph of $f (x) = x^{2}$ to sketch: (a) $g (x) = x^{2} - 1$ , (b) $h (x) = (x + 2)^{2}$ , (c) $F (x) = (x - 2)^{2}$ , (d) $G (x) = (x - 2)^{2} + 2$ .

Solution

(a)

g (x) = f (x) - 1

: shift the parabola down 1 unit.

Graph showing y=x² shifted down 1 to y=x²-1.

(b)

h (x) = f (x + 2)

: shift the parabola left 2 units. The vertex moves from $(0,0)$ to

(- 2, 0)

Graph showing y=x² shifted left 2 to y=(x+2)².

(c)

F (x) = f (x - 2)

: shift the parabola right 2 units. The vertex moves to $(2,0)$.

Graph showing y=x² shifted right 2 to y=(x-2)².

(d)

G (x) = F (x) + 2

: take the graph from (c) and shift it up 2 units. The vertex is now at $(2, 2)$.

Graph showing y=(x-2)² shifted up 2 to y=(x-2)²+2.

Reflection in the $x$ -Axis: $y = - f (x)$

The graph of $y = - f (x)$ is obtained by reflecting the graph of $y = f (x)$ in the $x$ -axis. Every point $(a, b)$ maps to $(a, - b)$ .

Diagram showing f(x) reflected across the x-axis to produce -f(x). — Reflection in the $x$ -axis negates all $y$ -values.

Reflection in the $y$ -Axis: $y = f (- x)$

The graph of $y = f (- x)$ is obtained by reflecting the graph of $y = f (x)$ in the $y$ -axis. Every point $(a, b)$ maps to $(- a, b)$ .

Diagram showing f(x) reflected across the y-axis to produce f(-x). — Reflection in the $y$ -axis negates all $x$ -values.

Vertical Scaling: $y = c f (x)$

If 
  c
  >
  1
: vertically stretch the graph of 
  f
 away from the 
  x
-axis by factor 
  c
.
If 
  0
  <
  c
  <
  1
: vertically compress the graph toward the 
  x
-axis by factor 
  c
.
If 
  c
  <
  0
: apply the stretch or compression, then reflect in the 
  x
-axis.

Every point 
  (
  a
  ,
  b
  )
 maps to 
  (
  a
  ,
  c
  b
  )
.

Horizontal Scaling: $y = f (c x)$

If 
  c
  >
  1
: horizontally compress the graph toward the 
  y
-axis by factor 
  1
  /
  c
.
If 
  0
  <
  c
  <
  1
: horizontally stretch the graph away from the 
  y
-axis by factor 
  1
  /
  c
.
If 
  c
  <
  0
: apply the stretch or compression, then reflect in the 
  y
-axis.

Every point 
  (
  a
  ,
  b
  )
 maps to 
  (
  a
  /
  c
  ,
  b
  )
.

Combining Transformations: $y = k f (a x - b) + h$

To graph $y = k f (a x - b) + h$ from the graph of $y = f (x)$ , apply transformations in this order:

Horizontal scaling by $1 / | a |$ (reflect in $y$ -axis if $a < 0$ ).
Horizontal shift by $b / a$ units.
Vertical scaling by $| k |$ (reflect in $x$ -axis if $k < 0$ ).
Vertical shift by $h$ units.

Use the graph of $y = \sqrt{x}$ to obtain the graph of $y = 2 - \sqrt{2 x + 4}$ and determine its domain and range.

Solution

Rewrite:

y = - \sqrt{2 (x + 2)} + 2

. Step 1:

y = \sqrt{2 x}

— horizontal compression of

y = \sqrt{x}

by factor

1 / 2

Step 2:

y = \sqrt{2 (x + 2)}

— shift left 2 units.

Graph of y=√(2(x+2)) after shifting left 2.

Step 3:

y = - \sqrt{2 (x + 2)}

— reflect in the

x

-axis.

Step 4:

y = - \sqrt{2 (x + 2)} + 2

— shift up 2 units.

Final graph of y = 2 - √(2x+4) after all transformations.

Domain:

[- 2, \infty)

. Range:

(- \infty, 2]

Special Transformations: $y = | f (x) |$ and $y = f (| x |)$

$y = | f (x) |$ : Reflect any part of the graph of $f$ that lies below the $x$ -axis up to above the $x$ -axis. The part already above stays unchanged.

$y = f (| x |)$ : Remove the part of the graph for $x < 0$ , keep the right half ( $x \geq 0$ ), and reflect it in the $y$ -axis to fill the left side. The result is always an even function.

Graph showing y = |f(x)| with the negative portion reflected above the x-axis. — To get $y = | f (x) |$ , reflect all portions of the graph below the $x$ -axis upward.

Graph showing y = f(|x|) with the left side replaced by a reflection of the right side. — To get $y = f (| x |)$ , keep the right half and reflect it across the $y$ -axis to create the left half.

Frequently Asked Questions

Why does

f (x - c)

shift right and

f (x + c)

shift left?

Think about it this way: to get the same output from

f (x - c)

as from

f

at 0, you need

x - c = 0

, i.e.,

x = c

. So the reference point moves right to

x = c

. Similarly,

f (x + c) = 0

requires

x = - c

, so the reference point moves left.

What is the order of operations for combining multiple transformations?

Always apply horizontal transformations (scaling, then shifting) before vertical ones (scaling, then shifting). Within horizontal, scale before shift. Within vertical, scale before shift. This matches the order in the formula

y = k f (a x - b) + h

Does reflecting across the x-axis change the domain?

No. Reflecting

y = f (x)

across the

x

-axis to get

y = - f (x)

uses the same

x

-values, so the domain is unchanged. The range changes: every

y

-value is negated.

y = f (| x |)

always even?

Yes. For any

g (x) = f (| x |)

, we have

g (- x) = f (| - x |) = f (| x |) = g (x)

, confirming that

g

is even.

Quick Reference

Vertical Shifts: $y = f (x) \pm c$

Horizontal Shifts: $y = f (x \mp c)$

Reflection in the $x$ -Axis: $y = - f (x)$

Reflection in the $y$ -Axis: $y = f (- x)$

Vertical Scaling: $y = c f (x)$

Horizontal Scaling: $y = f (c x)$

Combining Transformations: $y = k f (a x - b) + h$

Special Transformations: $y = | f (x) |$ and $y = f (| x |)$

Frequently Asked Questions

Quick Links

Social Media

Transformations of Functions

Quick Reference

Vertical Shifts: y = f ( x ) ± c

Horizontal Shifts: y = f ( x ∓ c )

Reflection in the x -Axis: y = − f ( x )

Reflection in the y -Axis: y = f ( − x )

Vertical Scaling: y = c f ( x )

Horizontal Scaling: y = f ( c x )

Combining Transformations: y = k f ( a x − b ) + h

Special Transformations: y = | f ( x ) | and y = f ( | x | )

Frequently Asked Questions

Quick Links

Social Media

Vertical Shifts: $y = f (x) \pm c$

Horizontal Shifts: $y = f (x \mp c)$

Reflection in the $x$ -Axis: $y = - f (x)$

Reflection in the $y$ -Axis: $y = f (- x)$

Vertical Scaling: $y = c f (x)$

Horizontal Scaling: $y = f (c x)$

Combining Transformations: $y = k f (a x - b) + h$

Special Transformations: $y = | f (x) |$ and $y = f (| x |)$