Interpreting Nonlinear Functions Flashcards

A ball is launched from a raised platform, and its height in feet above the ground is modeled by h(t) = −16t² + 96t + 112, where t is the number of seconds after launch. The ball strikes the ground exactly 7 seconds after launch. At what time, in seconds, does the ball reach its maximum height?

Answer: t = 3

The maximum of a downward-opening parabola occurs at the axis of symmetry, which lies midway between the two roots. Setting h(t) = 0: −16t² + 96t + 112 = 0 simplifies to t² − 6t − 7 = 0, giving (t − 7)(t + 1) = 0, so t = 7 or t = −1. The axis of symmetry is at t = (−1 + 7)/2 = 3. The common mistake is to take half of 7 (the landing time), but that ignores the second root at t = −1, which arises because the ball was launched from an elevated platform, not from ground level.

The table below shows selected values of a function f. x: 1, 2, 3, 4, 5 f(x): 6, 14, 26, 42, 62 Assuming f is a quadratic function, what is the value of f(6)?

Answer: 86

The first differences are 8, 12, 16, 20 — not constant, so f is not linear. The second differences are all 4 (constant), confirming f is quadratic. Using the system of equations with f(1)=6, f(2)=14, f(3)=26 yields f(x) = 2x² + 2x + 2. Therefore f(6) = 2(36) + 2(6) + 2 = 72 + 12 + 2 = 86. The trap answer 82 comes from naively extending the last observed difference of 20 linearly (62 + 20 = 82), which ignores that the differences themselves are growing by 4 each step — so the next difference is 24, giving 62 + 24 = 86.

The number of active users on a platform t months after launch is modeled by N(t) = 1,200 · 2^(t/4). Which of the following correctly describes the relationship between N(t) and N(t + 4) for any value of t?

Answer: N(t + 4) = 2 · N(t)

Computing the ratio: N(t + 4) / N(t) = [1200 · 2^((t+4)/4)] / [1200 · 2^(t/4)] = 2^((t+4)/4 − t/4) = 2^(4/4) = 2^1 = 2. So N(t + 4) = 2 · N(t) for any t — the user count doubles every 4 months regardless of the current value of t. Choice A is wrong because exponential growth is multiplicative, not additive. Choice B would imply quadrupling every 4 months. Choice D describes a different kind of growth entirely.

Let g(x) = x² − 8x + 15. If g(2) = g(n) for some integer n where n ≠ 2, what is the value of n?

Answer: 6

Rewriting in vertex form: g(x) = (x − 4)² − 1, so the axis of symmetry is x = 4. Because a parabola is symmetric about its axis, g(a) = g(b) whenever a and b are equidistant from the axis. The point x = 2 is 2 units to the left of x = 4, so the symmetric point is x = 4 + 2 = 6. Verification: g(2) = 4 − 16 + 15 = 3 and g(6) = 36 − 48 + 15 = 3. ✓ Choice A (n = 4) is the x-coordinate of the vertex, where g(4) = −1 ≠ 3. Choice D is wrong because the symmetry of any non-linear quadratic guarantees such a point exists.

A company's annual revenue R(t), in thousands of dollars, is modeled by R(t) = −2t² + 20t + 50, where t is the number of years since founding. Between which pair of consecutive years does the revenue increase by the greatest amount?

Answer: Years 0 and 1

The year-over-year increase is R(t + 1) − R(t). Expanding: R(t+1) − R(t) = −2(t+1)² + 20(t+1) + 50 − (−2t² + 20t + 50) = −4t + 18. This expression is a decreasing linear function of t, meaning the annual increase is greatest when t is smallest — at t = 0 (years 0 to 1), where the gain is 18 thousand dollars. Many students intuitively pick years 4–5 (the year before the revenue peak at t = 5), but that interval has only a 2-thousand-dollar increase. For a concave-down parabola, the steepest gains occur at the beginning of the increasing phase, not near the peak.

The function p(x) = x(x − 3)(x + 5) models the profit, in hundreds of dollars, from producing and selling x units of a product, where x must be a non-negative integer. For which value(s) of x is the profit equal to zero?

Answer: x = 0 and x = 3 only

Setting p(x) = 0 gives x = 0, x = 3, or x = −5 as mathematical solutions. However, because x represents the number of units produced and sold, x must be a non-negative integer. The root x = −5 is mathematically valid but has no meaning in this real-world context and must be excluded. Therefore, only x = 0 (no units produced, no profit) and x = 3 (break-even at 3 units) satisfy both the equation and the domain constraint. Choice D is a distractor that confuses the factor (x + 5) with a root of +5.