learn · audeac — Interactive IB Mathematics

Vertical translation

y = f(x) + k

k>0

shifts UP by

k

. Intuitive direction.

Horizontal translation

y = f(x - h)

h>0

shifts RIGHT by

h

(counter-intuitive: SUBTRACTING

h

moves right). The vector form is

\binom{h}{k}

.

Vertical stretch

y = a\,f(x)\;(a>0)

Stretches by factor

a

from

x

-axis.

a>1

⇒ further from x-axis.

0<a<1

⇒ closer (compression).

Horizontal stretch

y = f(bx)\;(b>0)

Stretches by factor

1/b

from

y

-axis (reciprocal!).

b>1

⇒ COMPRESSION;

0<b<1

⇒ STRETCH.

Reflect in $x$-axis

y = -f(x)

Reflect in $y$-axis

y = f(-x)

Combined transformation

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

a

vert stretch ·

b

horiz stretch ·

h

horiz shift ·

k

vert shift. Order matters when mixing inside and outside.

Modulus transformations

y=|f(x)|

reflects parts BELOW the

x

-axis ABOVE it.

y=f(|x|)

replaces the

x<0

part with a mirror image of the

x>0

part across the

y

-axis.

Reciprocal transformation

y = \dfrac{1}{f(x)}

Zeros of

f

⇒ vertical asymptotes of

1/f

. Where

f

has a min,

1/f

has a max (at the same

x

).

1/f

has the same sign as

f

.

Rational function

y = \dfrac{ax+b}{cx+d}

VA at

x=-d/c

; HA at

y=a/c

. Equivalent transformed form

y=\tfrac{A}{B(x-h)}+k

.

Reciprocal $1/x$

Odd, axes are asymptotes, self-inverse.

D = R = \mathbb{R}\setminus\{0\}

.

Radical (square root)

y = \sqrt{ax+b}

Domain:

ax+b\ge 0

. Range:

y\ge 0

(or

\le 0

if outer minus). Curve starts at boundary and rises with decreasing slope.

Absolute value

|a| = \begin{cases}a & a\ge 0\\ -a & a<0\end{cases}

Modulus equation

|A| = b\;\;\Rightarrow\;\; A = \pm b

Solve both cases and CHECK each against the original equation.

Modulus inequality (less than)

|A| \le b\;\;\Leftrightarrow\;\;-b \le A \le b

Modulus inequality (greater than)

|A| \ge b\;\;\Leftrightarrow\;\;A \le -b\;\text{ or }\;A \ge b

Asymptotes — vertical

Set denominator

=0

; the values are VAs (unless they cancel with the numerator).

Asymptotes — horizontal

Compare degrees: numerator deg

<

denominator deg ⇒

y=0

. Equal degrees ⇒

y=

leading-coeff ratio. Higher numerator ⇒ no HA (slant or beyond).

Pitfall — order of transformations

Apply inside-the-bracket transformations first (in reverse algebra order): translate inside ⇒ then stretch inside. Outside transformations follow in natural order.

Pitfall — sign of $h$

f(x-h)

shifts RIGHT by

h

,

f(x+h)

shifts LEFT by

h

. Inside the bracket the sign flips its apparent meaning.

Pitfall — extraneous solutions of $|A|=B$

If

B<0

there is no solution. If

B

contains

x

, always check — squaring or splitting can introduce false roots.