College Algebra / Unit 7 Alg 360 worked example, 60 to 90 seconds
Sketching a Polynomial
The idea
Sketching a polynomial means setting the arms by the leading term, planting the zeros with their multiplicities, marking the y-intercept, and drawing one smooth curve through the evidence.
In practice
Sketch f(x) = x³ − 3x + 2, which factors as (x − 1)²(x + 2). Arms first: leading term x³, odd degree, positive coefficient — left arm down, right arm up. Zeros: −2 crosses, 1 touches and turns, by their multiplicities. The y-intercept sits at (0, 2), read from the constant. Now connect: rise through −2, top out, come down to kiss the axis at 1, and climb away right. Audit the sketch: degree 3 allows at most two turning points, and the drawing used exactly two — no extra wiggles allowed, none drawn.
Where people slip
Smoothness is doctrine: polynomial graphs have no corners, breaks, or holes.
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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