Transformations Practice

Function notation for reflection across the y-axis

$$f(-x)$$

Function notation for reflection across the x-axis

$$-f(x)$$

Type of reflection when $x$ becomes negative inside $f(-x)$

Horizontal reflection across the y-axis

Type of reflection when $y$ becomes negative in $-f(x)$

Vertical reflection across the x-axis

Transformation represented by $f(2x)$

Horizontal compression by a factor of $\frac{1}{2}$

Transformation represented by $f\left(\frac{1}{2}x\right)$

Horizontal stretch by a factor of $2$

Horizontal scale factor formula for $f(bx)$

$$\frac{1}{|b|}$$

Transformation represented by $2f(x)$

Vertical stretch by a factor of $2$

Transformation represented by $\frac{1}{2}f(x)$

Vertical compression by a factor of $\frac{1}{2}$

Vertical scale factor formula for $af(x)$

$$|a|$$

Translation direction and distance for $f(x - 3)$

Shift right $3$ units

Translation direction and distance for $f(x + 3)$

Shift left $3$ units

Translation direction and distance for $f(x) + 4$

Shift up $4$ units

Translation direction and distance for $f(x) - 4$

Shift down $4$ units

Reason why $f(x - 5)$ shifts to the right instead of left

Horizontal translations act opposite to the sign inside the function.

Role of parameter $a$ in $y = af(b(x - h)) + k$

Controls vertical stretch/compression and reflection across the x-axis.

Role of parameter $b$ in $y = af(b(x - h)) + k$

Controls horizontal stretch/compression and reflection across the y-axis.

Role of parameter $h$ in $y = af(b(x - h)) + k$

Controls horizontal translation (left/right shift).

Role of parameter $k$ in $y = af(b(x - h)) + k$

Controls vertical translation (up/down shift).

New coordinates when $(6, 4)$ undergoes a horizontal compression by $\frac{1}{2}$

$$(3, 4)$$

New coordinates when $(3, 5)$ is reflected across the x-axis

$$(3, -5)$$

New coordinates when $(3, 5)$ is reflected across the y-axis

$$(-3, 5)$$

New coordinates when $(-2, 7)$ is shifted right $4$ units

$$(2, 7)$$

True or False: $f(3x)$ is a horizontal stretch by $3$

False. It is a horizontal compression by a factor of $\frac{1}{3}$.

True or False: $3f(x)$ is a vertical stretch by $3$

True.

Transformations in $g(x) = -2f(x)$

Vertical stretch by a factor of $2$ and reflection across the x-axis.

Transformations in $g(x) = f(x - 4) + 7$

Shift right $4$ units and shift up $7$ units.

Transformations in $g(x) = -f(x + 3) - 2$

Reflection across the x-axis, shift left $3$ units, and shift down $2$ units.

Transformations in $g(x) = 3f\left(\frac{1}{2}(x - 4)\right) + 2$

Vertical stretch by $3$, horizontal stretch by $2$, shift right $4$ units, and shift up $2$ units.

Rule for inside vs. outside transformations

Inside changes affect $x$ (horizontal transformations), outside changes affect $y$ (vertical transformations).