Sketching a Polynomial

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.

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.