ACT Preparation / Unit 11 ACT 1032 review, 2 to 3 minutes
A Ball in the Air
The idea
The unit's tools all describe curves and the equations that cut them. A quadratic equation is solved by the zero-product property after factoring, by the square root property, by completing the square, or by the quadratic formula, and the discriminant counts the real solutions before any are found. The roots sum to −b/a and multiply to c/a. Radical equations are solved by raising both sides to a power, and every answer is checked for extraneous solutions. A parabola opens up or down, with its vertex at x = −b/(2a), read directly from vertex form, and the vertex gives the maximum or minimum. Polynomials have end behavior set by the leading term and zeros whose multiplicity decides crossing or touching; the remainder and factor theorems tie zeros to factors. Rational functions have holes and asymptotes, and variation models are products or quotients with a constant k.
In practice
Worked example: in an invented model, a ball is thrown upward, and its height in feet after t seconds is h(t) = −16t² + 64t + 80. The parabola opens down, so its vertex is a maximum: t = −64/(2 × −16) = 2 seconds, and h(2) = −64 + 128 + 80 = 144 feet. When does it land? Set h(t) = 0 and divide by −16: t² − 4t − 5 = 0, which factors as (t − 5)(t + 1) = 0, so t = 5 or t = −1. Time cannot be negative, so the ball lands at 5 seconds. The roots' sum, 5 + (−1) = 4, matches −b/a for t² − 4t − 5, and their product, −5, matches c/a. The starting height is h(0) = 80 feet, the intercept.
Where people slip
Three habits carry forward. Check every answer against the situation and the original equation: a negative time, like an extraneous root, is discarded. Read which quantity a question asks for, the input at the vertex or the value there. And let the form of an expression choose the method. The next unit turns to growth that multiplies instead of adding: exponential functions and their inverses, the logarithms.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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