Intermediate Algebra | 33.3: The Leading Coefficient a
The coefficient a in f(x) = ax² is the single number that controls everything about a parabola's shape. This video builds both rules from scratch using a comparison table, then shows how to apply them to any quadratic — including negative values and fractions — with no graphing required.
Key concepts covered:
• The baseline parabola f(x) = x² and how it serves as a reference shape for comparison
• Sign rule: positive a opens the parabola upward; negative a flips it downward
• Width rule: absolute value of a greater than 1 produces a narrower curve; absolute value of a between 0 and 1 produces a wider curve
• Order of operations: square x first, then multiply by a — and why reversing this produces a wrong answer
• Why fraction form is misleading: a = 5/2 equals 2.5, which is greater than 1 and makes the parabola narrower, not wider
• Combining both rules to instantly read parabolas like −3x² (narrow, downward) and −(1/5)x² (wide, downward) without building a table
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SOURCE MATERIALS
The source materials for this video are from • Intermediate Algebra Lecture 11.5: Sketch...