Parabola Width: The a That Squeezes
A parabola's width is set by the leading coefficient a: larger |a| makes it narrower, smaller |a| makes it wider.
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What controls how wide or narrow a parabola is?
Two parabolas both touch the origin: y = x² and y = 3x². One looks like a gentle U, the other like a narrow V-shaped cup.
People often think a parabola's width comes from moving it left/right or changing the number at the end. But neither of those changes its shape—only the multiplier changes how fast it rises.
A side-by-side table and an interactive slider for a show the same x-values producing much larger y-values when a is larger.
Read the parabola's width directly from the leading coefficient: larger |a| means narrower, smaller |a| means wider.
- vertex form translations
- roots and factoring
- axis of symmetry
- conic sections
- 01Same Vertex, Different WidthslideSlot 1Hook
Show y=x² and y=3x² side by side.
- Both have vertex at (0, 0)
- Same vertex, different widths
- What causes it?
PhenomenonTwo parabolas with the exact same vertex can look very different: y=x² is wide, y=3x² is narrow.
QuestionIf the vertex isn't moving, what controls the width?
- 02Where We Expect the ChangeslideSlot 2Tension
Pose common guess.
- Common guess: vertex position or constant term controls width
- Prediction: shifting the graph should stretch it
- But shifting only slides the graph, it doesn't change its shape
PredictionMove the parabola left or right and it will widen.
Tempting intuitionWe focus on where the vertex is and expect location to change the graph's footprint.
- 03Slide a and Watch the Graph PinchinteractiveSlot 3Reveal
Use the slider to change a and observe width while vertex stays fixed.
- As a increases, y at x=1 grows from 1 to a larger value
- Same x, bigger y means steeper arms
- Steeper arms close in sooner, making the parabola narrower
EvidenceAt x=1, y=1 for y=x², y=2 for y=2x², and y=0.5 for y=0.5x²; at every fixed x, the height is multiplied by a.
ConclusionThe width is determined by a, the multiplier on x².
Mechanism- 1For y = ax², height at a fixed horizontal distance from the vertex is a times x².
- 2Bigger a makes height grow faster as x moves away, so arms rise more steeply.
- 3Steeper arms reach the same height at a smaller x, so the graph is narrower.
- 04Read the Width from aslideSlot 4Takeaway
Transfer the rule to a new scenario.
- Larger |a| → narrower parabola
- Smaller |a| → wider parabola
- Vertex shifts don't change width
TransferImagine y = 10x² compared with y = 0.1x²: even without graphing, you can predict that 10x² will be tight and narrow, while 0.1x² will spread out wide.
Expected inferenceWhen comparing parabolas in vertex form, ignore the h and k for width—only compare the leading multiplier a to decide which is narrower.
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