MathsFrancais2 min de lectureMis a jour 2 avr. 2026

Visualizing Function Transformations

Master the rules of shifts, stretches, and reflections. Learn how mathematical modifications inside and outside a function change its position and shape on the coordinate plane.

Points Cles

Vertical Shifts occur when a constant is added or subtracted outside the function.
Horizontal Shifts occur inside the function and move in the opposite direction of the sign.
Reflections over the x-axis are caused by a negative sign in front of the function.
Vertical Stretch (|a|>1) makes the graph narrower; Compression (|a|<1) makes it wider.
Transformations follow a specific order of operations: Stretches, then Reflections, then Shifts.

Parent Functions: The Basic Building Blocks

In algebra, we start with "Parent Functions"—the simplest version of a function family. For example, f(x) = x² is the parent of all parabolas. A transformation is any change that moves, flips, or distorts that parent shape into a new position or size. Think of a parent function as a template, and transformations as the adjustments you make to fit that template to a specific set of real-world data.

The Transformation Matrix: Quick Rules

Use this matrix to identify exactly how a function will move based on its equation. For the rules below, assume **c > 0**. ``` | Modification | Transformation Type | Practical Movement | | --- | --- | --- | | f(x) + c | Vertical Shift | Move UP by c units | | f(x) - c | Vertical Shift | Move DOWN by c units | | f(x - c) | Horizontal Shift | Move RIGHT by c units | | f(x + c) | Horizontal Shift | Move LEFT by c units | | -f(x) | Reflection | Flip over the X-axis | | f(-x) | Reflection | Flip over the Y-axis | ```

Dilations: Stretching and Compressing

A dilation changes the "intensity" of the function's curve. When you multiply the function by a constant (a), the y-values are scaled. - **Vertical Stretch**: If |a| > 1, the graph grows faster and looks "thinner." - **Vertical Compression**: If 0 < |a| < 1, the graph grows slower and looks "wider" or flatter.

Step-by-Step: Transforming f(x) = x²

Let's analyze the transformed function: **g(x) = -2(x - 3)² + 5** ``` 1. Parent: Start with the parabola f(x) = x². 2. Stretch: The "2" stretches it vertically by 2. 3. Reflect: The "-" flips it over the x-axis (now it opens down). 4. Horizontal Shift: The "(x - 3)" moves it 3 units RIGHT. 5. Vertical Shift: The "+ 5" moves it 5 units UP. ``` Result: The vertex is now at (3, 5) and the parabola opens downward.

The Order of Operations

When a function has multiple transformations, the order in which you apply them matters. A common mistake is shifting before reflecting, which can result in the wrong final position. In general, follow the order of **Scale -> Reflect -> Translate** (SRT).