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.
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.
| 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 |
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.
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).
This is counter-intuitive! It happens because you are changing the "input" value. To get the same output as the parent function at x=0, you now need to input x=-3. Thus, the entire graph shifts 3 units toward the negative side.
For a linear function (like y = mx), yes. For other functions (like x²), a negative sign reflects the entire shape without changing the fundamental growth rate of the variable.
Yes. If you multiply the x *inside* the function, e.g., f(bx), it creates a horizontal dilation. This is less common in standard algebra but very common in trigonometry.
In a parabola, the vertex (h, k) moves exactly according to the shifts. A horizontal shift of +3 moves the vertex to x=3, and a vertical shift of +5 moves it to y=5.
Vertical shifts and reflections directly change the range (y-values). Horizontal shifts change where certain values appear but usually don't change the overall domain of a real-valued polynomial.
Translations and Reflections are "Rigid" because they do not change the size or shape of the graph, only its location. Dilations (stretches) are "Non-Rigid."
The best way is to pick an easy point (like x=0 or the old vertex) and plug it into the new equation to see if the calculated y-value matches your graph.
