The Effect of a on the Graph of y = ax²
Simple Explanation
In y = ax², the coefficient a controls two things: its sign determines whether the parabola opens upward (a > 0) or downward (a < 0), and its size determines how narrow or wide the parabola is — a larger |a| makes it narrower (steeper); a smaller |a| makes it wider (flatter).
Why Do We Need It?
Understanding a's role lets you sketch a rough graph of any y = ax² instantly, just by reading its coefficient, before plotting a single point.
See It
A wide upward-opening parabola, y = 0.5x squared, showing a flatter curve than the standard y = x squared
Worked Example
Compare the width of two parabolas
Which is narrower: y = 3x² or y = 0.5x²?
Why Does This Work?
For any fixed x-value (other than 0), a larger |a| multiplies that same squared value by a bigger number, producing a larger y — pulling the curve closer to the vertical axis, i.e. making it visually narrower.
Real-Life Example
Satellite dish and headlight reflector shapes
Parabolic reflectors (satellite dishes, car headlights, flashlight mirrors) are designed with a specific "narrowness" to focus signals or light at one exact point.
Engineers choose the coefficient a precisely to control how narrow or wide the parabolic reflector's cross-section is, which determines where it focuses.
Practice
Which parabola opens downward and is narrower than y = x²?
MediumCommon mistake
Confusing the effect of the sign of a (direction) with the effect of its size (width) — they are two completely separate properties that must be checked independently.
Quick Review
- Sign of a: positive opens upward; negative opens downward.
- Size of a: |a| > 1 makes the parabola narrower; 0 < |a| < 1 makes it wider.
- y = x² is the "standard" width to compare every other parabola against.