Vertical Stretches, Compressions, and Reflections of Piecewise Functions
Vertical stretches, compressions, and reflections of piecewise functions is a Grade 11 Algebra 1 topic from enVision Chapter 5 applying the coefficient a in g(x) = a * f(x) to transform graphs. For the absolute value function |x|: g(x) = 3|x| stretches the V-shape vertically by 3, making it narrower. h(x) = (1/2)|x| compresses it vertically, making the V wider since each y-value is halved. k(x) = -2|x| both reflects the graph downward and stretches it. The coefficient a multiplies every output value — it does not move the vertex but changes how steeply the graph rises.
Key Concepts
The coefficient $a$ in the transformation $g(x) = a \cdot f(x)$ affects the graph of any function $f(x)$ by stretching, compressing, or reflecting it vertically. If $0 < |a| < 1$, the graph is compressed vertically (appears wider for functions like $|x|$) If $|a| 1$, the graph is stretched vertically (appears narrower for functions like $|x|$) If $a < 0$, the graph is reflected across the x axis.
Common Questions
What does g(x) = 3|x| look like compared to |x|?
It is vertically stretched by 3. The V-shape is narrower because y-values are tripled, causing the graph to rise steeply. The vertex remains at (0,0).
What does h(x) = (1/2)|x| look like compared to |x|?
It is vertically compressed. Each y-value is halved, making the V-shape wider. The vertex stays at (0,0).
What does k(x) = -2|x| do to the graph?
It stretches by 2 and reflects across the x-axis. The V opens downward and is narrower than the parent.
Does the coefficient a affect the vertex location?
No. The vertex stays at the same (h, k) position regardless of a. The coefficient only changes the steepness and direction.
When does a vertical transformation produce a compression vs a stretch?
Compression when 0 < |a| < 1 (y-values shrink toward x-axis). Stretch when |a| > 1 (y-values move away from x-axis).
How does a negative coefficient affect any piecewise function?
It reflects the entire graph across the x-axis. Every positive output becomes negative and vice versa.