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Transformations

Stretch, reflect, translate — every shape from one parent.

Interactive graph

Interactive Lab

Drag corner to resize ↘

Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualThe simplest member of a family. Everything else is a reshaped version of it.

AlgebraStart with $f(x)$ . All transformations are applied to this base.

VisualPull the graph taller (|a|>1) or squash it flatter (0<|a|<1). Negative a flips it across the x-axis.

AlgebraMultiplies every output by $a$ . The shape stretches vertically.

y = a\cdot f(x)

VisualSqueeze the graph toward the y-axis (|k|>1) or pull it wider (0<|k|<1). Negative k reflects across the y-axis.

AlgebraReplaces every $x$ with $kx$ . Note: the effect on width is inverse to $k$ .

y = f(kx)

Visualh shifts right (when positive), v shifts up. Both move the whole graph without distortion.

AlgebraSubtract $h$ inside the function (right shift), add $v$ outside (up shift).

y = f(x-h) + v

General transformation

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

All four transformations at once

Reflection in x-axis

y = -f(x)

a = -1

Reflection in y-axis

y = f(-x)

k = -1

Mark this lesson complete to update your mastery.

easytransformations

The graph of

y=f(x)+3

is the graph of

f

shifted how?

easytransformations

Describe

y = -f(x)

in words.

mediumtransformations

The graph of

y=f(x-2)+5

f

shifted:

mediumtransformations

f(x)=x^2

, write the equation of

f

stretched vertically by 2 and shifted left 3.

hardtransformations

f(x)=\sqrt{x}

, write

y=2f(3x-6)-4

in the form

a\,f\big(k(x-h)\big)+v