Graph transformations (translations, stretches, reflections)

1. Vertical and horizontal shifts: $f(x) \pm k$ and $f(x \pm k)$

Vertical and Horizontal Shifts

A vertical shift moves the graph of $f(x)$ up or down by adding or subtracting a constant outside the function: $f(x) + k$ shifts up by $k$ units when $k > 0$, and $f(x) - k$ shifts down by $k$ units. A horizontal shift moves the graph left or right by modifying the input: $f(x - h)$ shifts right by $h$ units when $h > 0$, and $f(x + h)$ shifts left by $h$ units.

Vertical shifts affect output values directly, while horizontal shifts affect input values before function evaluation.

Core Rules:

• $f(x) + k$: shift graph up $k$ units (add to outputs)
• $f(x) - k$: shift graph down $k$ units (subtract from outputs)
• $f(x - h)$: shift graph right $h$ units (replace $x$ with $x - h$)
• $f(x + h)$: shift graph left $h$ units (replace $x$ with $x + h$)

These transformations preserve the shape of the graph; only position changes.

Example: If $f(x) = x^2$ and we form $g(x) = (x - 3)^2 + 2$, the vertex moves from $(0, 0)$ to $(3, 2)$.

2. Reflections across the $x$-axis and $y$-axis

Reflections Across Axes

A reflection across the $x$-axis replaces $f(x)$ with $-f(x)$, flipping the graph vertically by negating all output values. A reflection across the $y$-axis replaces $f(x)$ with $f(-x)$, flipping the graph horizontally by negating all input values before evaluation.

Reflections reverse orientation along one axis while preserving distances from that axis.

Core Rules:

• $-f(x)$: reflect across the $x$-axis (multiply outputs by $-1$)
• $f(-x)$: reflect across the $y$-axis (replace $x$ with $-x$ in inputs)
• Reflections preserve shape and size but reverse direction
• Combining both gives $-f(-x)$, equivalent to a 180-degree rotation about the origin

These transformations are their own inverses: applying the same reflection twice returns the original graph.

Example: If $f(x) = \sqrt{x}$ for $x \geq 0$, then $-f(x) = -\sqrt{x}$ points downward, and $f(-x) = \sqrt{-x}$ is defined for $x \leq 0$.

3. Vertical and horizontal stretches and compressions

Vertical and Horizontal Stretches and Compressions

A vertical stretch multiplies outputs by a factor $a > 1$ in $a \cdot f(x)$, pulling the graph away from the $x$-axis; when $0 < a < 1$, it is a vertical compression toward the $x$-axis. A horizontal stretch occurs with $f(bx)$ when $0 < b < 1$, spreading the graph away from the $y$-axis; when $b > 1$, it is a horizontal compression toward the $y$-axis.

Vertical scaling affects $y$-coordinates, while horizontal scaling affects $x$-coordinates inversely.

Core Rules:

• $a \cdot f(x)$ with $a > 1$: vertical stretch by factor $a$
• $a \cdot f(x)$ with $0 < a < 1$: vertical compression by factor $a$
• $f(bx)$ with $b > 1$: horizontal compression by factor $\frac{1}{b}$
• $f(bx)$ with $0 < b < 1$: horizontal stretch by factor $\frac{1}{b}$

Horizontal scaling is counterintuitive: larger $b$ compresses the graph.

Example: $2\sin(x)$ has amplitude 2 (vertical stretch), while $\sin(2x)$ has period $\pi$ (horizontal compression by $\frac{1}{2}$).

4. Sequencing multiple transformations and their effect on coordinate pairs

Sequencing Multiple Transformations

When multiple transformations are applied, the order matters for operations inside versus outside the function. Transformations inside $f(...)$ (horizontal shifts, stretches, reflections) are applied to $x$-coordinates in reverse order of algebraic operations. Transformations outside (vertical shifts, stretches, reflections) are applied to $y$-coordinates in standard order.

A point $(x_0, y_0)$ on $f(x)$ transforms systematically through each operation.

Core Rules:

• Apply horizontal transformations first to $x$-coordinates (right-to-left inside function)
• Then apply vertical transformations to $y$-coordinates (left-to-right outside function)
• For $a \cdot f(b(x - h)) + k$: shift right $h$, compress by $\frac{1}{b}$, stretch by $a$, shift up $k$
• Track coordinate pairs: $(x_0, y_0) \to (x_0 + h, y_0) \to (\frac{x_0 + h}{b}, y_0) \to (\frac{x_0 + h}{b}, a \cdot y_0) \to (\frac{x_0 + h}{b}, a \cdot y_0 + k)$

Example: For $2f(3(x - 1)) + 4$, point $(0, 1)$ becomes $(1, 1) \to (\frac{1}{3}, 1) \to (\frac{1}{3}, 2) \to (\frac{1}{3}, 6)$.

5. Applications: Adjusting signal waveforms in telecommunications or fitting models to shifted experimental data

Applications in Signal Processing and Data Modeling

Graph transformations model real-world adjustments to periodic signals and experimental datasets. In telecommunications, vertical stretches adjust signal amplitude (power), horizontal compressions modify frequency (data rate), and horizontal shifts introduce phase delays for synchronization. In experimental science, transformations align theoretical models with observed data by shifting baselines or rescaling measurements.

These operations preserve functional form while matching physical constraints.

Core Rules:

• Amplitude modulation: vertical stretch $A \cdot \sin(\omega t)$ controls signal strength
• Frequency modulation: horizontal compression $\sin(\omega t)$ with $\omega > 1$ increases oscillation rate
• Phase shift: horizontal shift $\sin(\omega(t - \phi))$ delays waveform by $\phi$ seconds
• Baseline correction: vertical shift $f(t) + C$ adjusts zero-level in measurements

Combining transformations enables precise signal design and model calibration.

Example: A carrier wave $\sin(2\pi \cdot 1000 t)$ at 1000 Hz shifted by 0.001 seconds becomes $\sin(2\pi \cdot 1000(t - 0.001))$, introducing a phase lag.