2.6 Transformations of Functions

1. Vertical Transformations

Vertical manipulations alter the outputs ($y$-values) and occur strictly "outside" the primary function evaluation. Assume $a > 0$ and $b > 1$.

Notation Geometric Transformation
$f(x) + a$Vertical translation $a$ units up
$f(x) - a$Vertical translation $a$ units down
$b \cdot f(x)$Vertical stretch by a scale factor of $b$
$f(x) / b$Vertical stretch by a scale factor of $1/b$ (shrink)
$-f(x)$Reflection across the x-axis

2. Horizontal Transformations

Horizontal manipulations alter the inputs ($x$-values) and occur strictly "inside" the function bracket. These operations act counter-intuitively.

Notation Geometric Transformation
$f(x + a)$Horizontal translation $a$ units to the left
$f(x - a)$Horizontal translation $a$ units to the right
$f(bx)$Horizontal stretch by a scale factor of $1/b$ (shrink)
$f(x/b)$Horizontal stretch by a scale factor of $b$
$f(-x)$Reflection across the y-axis

EXAMPLE 2 (Point Translations)

Let $A(6, 10)$ be a coordinate situated on the curve $y = f(x)$. Applying discrete transformations yields new coordinate pairs $A'$:

$f(x) + 2 \Rightarrow A'(6, 12)$ (moves 2 up)
$2f(x) \Rightarrow A'(6, 20)$ (y multiplied by 2)
$-f(x) \Rightarrow A'(6, -10)$ (y coordinate negated)
$f(x - 2) \Rightarrow A'(8, 10)$ (moves 2 right, counter-intuitive)
$f(2x) \Rightarrow A'(3, 10)$ (x divided by 2, shrink)
$f(-x) \Rightarrow A'(-6, 10)$ (x coordinate negated)

3. Composite Transformations

Vector Notation: A horizontal translation $a$ and vertical translation $b$ combined can be strictly defined utilizing the translation vector $\binom{a}{b}$. For the quadratic vertex form $y=a(x-h)^2+k$, the base parabola shifts by the translation vector $\binom{h}{k}$.

Order of Execution: The specific order applied to composite sequences dictates algebraic accuracy.

  • For vertical $y = 2f(x) + 3$, stretch by 2 first, then translate up by 3. Reversing the execution creates $2[f(x) + 3] = 2f(x) + 6$.
  • For horizontal $y = f(2x + 6)$, evaluate input changes relative to $x$. Expressing as $f(2(x+3))$ accurately reveals a horizontal shrink by factor $1/2$, followed by a translation left by 3.