AHL 2.8 — TRANSFORMATIONS OF GRAPHS

📌 Understanding Transformations

• The base function y = f(x) represents the original untransformed graph, which acts as the reference shape before any movement, stretching, shrinking, or reflection is applied to it.
• A transformation is a mathematical operation that moves or reshapes a graph while preserving its fundamental structure, such as translations, reflections, and stretches.
• The coordinate axes remain fixed and invariant during transformations, meaning all movements are measured relative to these stationary reference lines.
• Translations are often written using vector notation (a, b), where a controls horizontal motion and b controls vertical motion of the entire graph.

📌 Translations — Shifting Graphs

• The transformation y = f(x) + b moves every point on the graph vertically by b units, upward if b is positive and downward if b is negative.
• Example: y = x² + 4 moves the parabola four units upward without altering its width or orientation.
• The transformation y = f(x − a) moves the graph horizontally, shifting it right by a units if a is positive and left by a units if a is negative.
• Example: y = √(x − 2) shifts the square-root graph two units to the right.
• The combined transformation y = f(x − a) + b applies both shifts using the translation vector (a, b).
• Example: y = (x − 3)² − 2 shifts the parabola three units right and two units downward.

📌 Reflections — Flipping Graphs

• The graph y = −f(x) reflects the original curve across the x-axis, reversing all vertical values while preserving horizontal positions.
• Example: y = x² becomes y = −x², changing an upward-opening parabola into a downward-opening one.
• The transformation y = f(−x) reflects the graph across the y-axis, producing a horizontal mirror image of the original curve.
• Example: y = eˣ becomes y = e⁻ˣ, reversing its growth direction while maintaining exponential behavior.

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📌 Stretches — Scaling Graphs

• Vertical stretches are written as y = p·f(x), multiplying all y-values by p, which changes amplitude or vertical height without altering the x-structure.
• Example: y = 2 sin x doubles the wave’s amplitude while keeping the same period.
• Horizontal stretches take the form y = f(qx), compressing the graph horizontally when q > 1 and stretching it when 0 < q < 1.
• Example: y = sin(2x) halves the original period of the sine curve.

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📌 Composite Transformations — Multiple Changes

• Composite transformations involve applying more than one transformation sequentially, producing graphs that combine stretching, reflection, and translation together.
• The order of transformations is critical: horizontal changes inside f( ) must always be applied before vertical transformations outside f( ).
• Example: y = x² → y = 3x² applies a vertical stretch of factor 3 → y = 3x² + 2 shifts the graph upward two units.
• Example: y = sin x → y = 4 sin(2x) doubles frequency and quadruples amplitude.

📌 Common Misconceptions

• Horizontal transformations behave opposite to intuition, meaning y = f(x − 2) shifts right rather than left.
• Vertical transformations behave intuitively, meaning y = f(x) + 3 always shifts upward by three units.
• Altering the order of composite transformations can completely change the final position and shape of the graph.