Vertical Transformations of Absolute Value Functions
The transformation f(x) = a|x| vertically stretches or compresses the absolute value parent function by factor |a|, and a negative a reflects the graph across the x-axis. In Grade 11 enVision Algebra 1 (Chapter 5: Piecewise Functions), students learn that when |a| > 1 the V-shape becomes narrower (vertical stretch), when 0 < |a| < 1 it becomes wider (vertical compression), and the vertex always remains at (0, 0). A negative coefficient a opens the V downward while preserving the vertex location.
Key Concepts
The transformation $f(x) = a|x|$ vertically stretches or compresses the parent function $f(x) = |x|$ by factor $|a|$. When $a 0$, the graph opens upward with vertex at $(0,0)$. When $a < 0$, the graph reflects across the x axis and opens downward with vertex at $(0,0)$.
Common Questions
What does the parameter a control in f(x) = a|x|?
The parameter a controls vertical stretch/compression (how wide or narrow the V-shape is) and reflection (whether the graph opens up or down).
What happens when |a| > 1 in f(x) = a|x|?
The graph stretches vertically, making the V narrower — the slopes of the two rays increase in steepness.
What happens when 0 < |a| < 1 in f(x) = a|x|?
The graph compresses vertically, making the V wider — the slopes of the two rays decrease in steepness.
What does a negative value of a do to the graph?
A negative a reflects the graph across the x-axis, making the V open downward instead of upward.
Does the vertex move when a changes in f(x) = a|x|?
No. The vertex remains at (0, 0) for f(x) = a|x| regardless of the value of a (only translations change the vertex position).
What is the graph of f(x) = −3|x|?
It is a downward-opening V-shape (reflected), narrower than the parent function (stretched by factor 3), with vertex at (0, 0).