College Algebra / Unit 6 Alg 335 worked example, 60 to 90 seconds
The Full Quadratic Workup
The idea
The full quadratic workup reads every feature from the formula: opening direction from a, vertex and axis from the vertex formula, y-intercept from c, x-intercepts from the zeros, extreme value from the vertex.
In practice
Work up f(x) = x² − 4x + 3. Direction: a = 1, opens up. Vertex: h = 2, k = f(2) = −1 — so (2, −1), axis x = 2, and the extreme is a minimum of −1. The y-intercept is (0, 3), read straight from c. The x-intercepts come from the zeros: the discriminant 16 − 12 = 4 promises two, and factoring delivers 1 and 3 — points (1, 0) and (3, 0). Five features, one formula, and every one checkable: the intercepts sit symmetric about the axis, 1 and 3 straddling x = 2 evenly, which is the geometry auditing the algebra.
Where people slip
The symmetry check is free; intercepts that straddle the axis unevenly announce an arithmetic slip.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
Builds on
- Nothing. This is a starting point.
Unlocks
- Nothing yet depends on this.
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