Interpreting Nonlinear Functions Flashcards

A company models its monthly profit, in thousands of dollars, with the function P(t) = -2t² + 16t - 24, where t is the number of months after launch. An analyst claims the company first becomes profitable at t = 2 months. Which of the following best evaluates this claim?

Answer: The claim is incorrect; the company first becomes profitable at t = 3 months, since P(3) = 6 > 0 and P(2) = 0 means the company breaks even at t = 2.

Evaluating P(2) = -2(4) + 16(2) - 24 = -8 + 32 - 24 = 0, which means the company breaks even — it does not yet have positive profit. P(3) = -2(9) + 48 - 24 = -18 + 48 - 24 = 6 > 0, so the company first earns positive profit at t = 3. The zero of P at t = 2 marks break-even, not profitability.

The function f(x) = 3(2^x) represents the number of bacteria in a culture after x hours. The function g(x) = 3(2^(x-4)) represents a second culture. Which statement correctly interprets the relationship between f and g?

Answer: Culture g reaches the same population as culture f, but exactly 4 hours later.

g(x) = 3(2^(x-4)) = f(x-4), which is a horizontal shift of f right by 4 units. This means g(x) equals the value that f had 4 hours earlier, so g reaches any population milestone exactly 4 hours after f does. Both cultures have the same growth rate (doubling every hour) and g(0) = 3(2^(-4)) = 3/16, which is much smaller than f(0) = 3, so culture g starts with fewer bacteria — but the key relationship is the 4-hour lag in reaching the same population level.

The height h (in feet) of a ball thrown upward is modeled by h(t) = -16t² + 64t + 5, where t is seconds. Which of the following is the most precise interpretation of the average rate of change of h on the interval [1, 3]?

Answer: The ball's height increases by an average of 0 feet per second over the interval, meaning it was at the same height at t = 1 and t = 3.

The average rate of change = [h(3) - h(1)] / (3-1). h(1) = -16+64+5 = 53 and h(3) = -144+192+5 = 53. So the average rate of change = (53-53)/2 = 0 ft/sec. This means the ball returned to the same height at t=3 as it was at t=1. Option C describes a true fact (the vertex is at t=2 so instantaneous velocity is 0 there), but that is NOT what the average rate of change measures — it's a property of the instantaneous rate at t=2, not the average over [1,3].

The function p(x) = x³ - 6x² + 9x is graphed in the xy-plane. For which of the following reasons is x = 3 NOT a zero at which the graph crosses the x-axis?

Answer: x = 3 is a zero of multiplicity 2, so the graph touches the x-axis at x = 3 but does not cross it.

Factoring: p(x) = x(x² - 6x + 9) = x(x-3)². The factor (x-3) appears with exponent 2 (multiplicity 2). When a zero has even multiplicity, the graph touches the x-axis at that point and bounces back — it does not cross. So at x = 3, p touches but does not cross the x-axis. At x = 0 (multiplicity 1, odd), the graph does cross. Option D is a misconception — multiplicity is determined by the factored form of p(x), not p'(x).

A population of deer is modeled by D(t) = 500 / (1 + 4e^(-0.3t)), where t is years since 2010. An ecologist states: 'The deer population will eventually stabilize near 500.' A second ecologist counters: 'The population grew fastest around t ≈ 4.6 years.' Which of the following is true?

Answer: Both ecologists are correct; the carrying capacity is 500 and the inflection point of this logistic model occurs where the population equals 250, which is near t ≈ 4.6.

This is a logistic function of the form D(t) = L/(1+Ae^(-kt)). The carrying capacity L = 500, confirming the first ecologist. The inflection point (fastest growth) of a logistic function occurs when D = L/2 = 250. Setting 500/(1+4e^(-0.3t)) = 250 gives 1+4e^(-0.3t) = 2, so 4e^(-0.3t) = 1, e^(-0.3t) = 0.25, -0.3t = ln(0.25) ≈ -1.386, t ≈ 4.62 years. Both ecologists are correct.

The graph of y = f(x) is a parabola with vertex at (2, 8) and passes through (0, 0). If g(x) = f(x + 3) - 5, what are the coordinates of the vertex of g, and what is g(0)?

Answer: Vertex of g is at (-1, 3) and g(0) = f(3) - 5.

Shifting f right by 3 and up by 5... wait — g(x) = f(x+3) - 5 shifts f LEFT by 3 and DOWN by 5. The vertex of f is (2, 8), so the vertex of g is at (2-3, 8-5) = (-1, 3). For g(0): g(0) = f(0+3) - 5 = f(3) - 5. We know f has vertex (2,8) and f(0) = 0, giving f(x) = -2(x-2)² + 8. So f(3) = -2(1) + 8 = 6, and g(0) = 6 - 5 = 1. However, option C states g(0) = f(3) - 5, which is the correct algebraic form before evaluating — and it correctly identifies the vertex. Option A states g(0) = -7, which is wrong. Option D states g(0) = -3, also wrong. Only C correctly states the vertex AND the correct unevaluated form of g(0), making it the most precisely correct answer without arithmetic errors from incorrect assumptions about f.