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Boss lesson · The Apex
Match point, exam-hall rules
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Match point, exam-hall rules

This is the boss. No calculator — today g=10 m/s2g = 10\ \mathrm{m/s^2}, stated on the paper, and every draw is chosen to fit in your head. One ball, one long problem, and every answer feeds the next: the apex time (that is the VERTEX of h(t)=v0t−5t2h(t) = v_0 t - 5t^2), then the height it buys, then the hang time (that is a ROOT), and finally Vieta signs the audit. Write each line down — one early slip rides the parabola all the way back to the ground.

Every tool from the chapter plays: vertex, roots, and the sum-of-roots check closing the loop. 5★ under par here is the chapter crown. Serve.

h

h=−b2ah = -\frac{b}{2a}

  • hh= Vertex x-coordinate
  • aa= Coefficient of x²
  • bb= Coefficient of x
Vertex x-Coordinate of a Parabola solver →
k

k=c−b24ak = c - \frac{b^2}{4a}

  • kk= Vertex y-coordinate
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Vertex y-Coordinate of a Parabola solver →
θv0H

H=v02sin⁡2θ2gH = \frac{v_0^{2} \sin^{2}\theta}{2g}

  • HH= Maximum height (length)
  • v0v_0= Launch speed (speed)
  • θ\theta= Launch angle (angle)
Projectile Maximum Height solver →
θv0T

T=2v0sin⁡θgT = \frac{2 v_0 \sin\theta}{g}

  • TT= Time of flight (time)
  • v0v_0= Launch speed (speed)
  • θ\theta= Launch angle (angle)
Projectile Time of Flight solver →
x

x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

  • xx= Root (plus branch)
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Quadratic Formula (Positive Root) solver →
no calculator in the boss room — the numbers are chosen to fit in your head