Comparing Function Types - Math 2

A ball is thrown upward with height function $h(t) = -16t^2 + 64t + 5$ , while a balloon rises with height function $b(t) = 5(1.2)^t$ , and an elevator moves with height function $e(t) = 5 + 8t$ . Which statement correctly describes when and why the balloon will be highest?
Answer: The balloon will eventually be highest because exponential functions with base greater than 1 increase without bound over time. The ball follows a quadratic (parabolic) path that reaches a maximum then falls. The elevator rises linearly. The balloon grows exponentially. Since exponential functions with base > 1 eventually exceed all polynomial functions, the balloon will eventually be highest and stay highest.
Three functions model population growth in different cities: $P_1(t) = 50000 + 1200t$ , $P_2(t) = 45000(1.03)^t$ , and $P_3(t) = 40000 + 800t + 20t^2$ . If current trends continue for 30 years, which city will have the largest population and why?
Answer: City 2 because exponential growth at 3% annually compounds to exceed both linear and quadratic growth over 30 years. After 30 years: P₁(30) = 86,000; P₂(30) = 45000(1.03)³⁰ ≈ 109,272; P₃(30) = 40000 + 24000 + 18000 = 82,000. The exponential function with 3% growth compounds to the largest value despite not having the largest initial population.
Two functions are defined: $p(x) = ax^2 + bx + c$ and $q(x) = d \cdot r^x$ , where $a, b, c, d, r > 0$ . Both functions pass through the points $(0, 3)$ and $(2, 12)$ . In the long run (as $x \to \infty$ ), which function will dominate, and what does this tell us about the fundamental difference between these function types?
Answer: Function $q(x)$ will dominate because exponential functions with base $r > 1$ eventually grow faster than any polynomial function, regardless of coefficients From the given points: $q(0) = d = 3$ and $q(2) = 3r^2 = 12$ , so $r^2 = 4$ and $r = 2$ . Thus $q(x) = 3 \cdot 2^x$ . For $p(x)$ : $p(0) = c = 3$ and $p(2) = 4a + 2b + 3 = 12$ , so $4a + 2b = 9$ . Regardless of specific values of $a$ and $b$ , exponential functions with base $> 1$ always eventually dominate polynomial functions. This illustrates the fundamental principle that exponential growth surpasses polynomial growth.
A population study tracks three different species over time. Species A follows the model $P_A(t) = 100 + 15t$ , Species B follows $P_B(t) = 80 + t^2$ , and Species C follows $P_C(t) = 50 \cdot 1.2^t$ . If the carrying capacity of the environment is 500 individuals, which species will be the first to approach this limit, and approximately when will this occur?
Answer: Species C will reach the limit first at approximately $t = 12.8$ because exponential growth eventually surpasses both linear and quadratic growth Solving each equation for when population reaches 500: Species A: $500 = 100 + 15t \Rightarrow t = 26.7$ . Species B: $500 = 80 + t^2 \Rightarrow t^2 = 420 \Rightarrow t ≈ 20.5$ . Species C: $500 = 50 \cdot 1.2^t \Rightarrow 1.2^t = 10 \Rightarrow t = \frac{\ln(10)}{\ln(1.2)} ≈ 12.6$ . Despite starting with the smallest initial population, the exponential function grows fastest and reaches the limit first.
Three functions are defined as follows: $f(x) = 2x + 3$ , $g(x) = x^2 - 1$ , and $h(x) = 3 \cdot 2^x$ . Over the interval $[0, 10]$ , which statement best describes the relative growth rates of these functions?
Answer: Function $h$ grows slowest initially, but its exponential nature causes it to eventually grow faster than both linear and quadratic functions At $x = 0$ : $f(0) = 3$ , $g(0) = -1$ , $h(0) = 3$ . Initially, the linear function $f$ has a constant rate of change of 2, while $h$ grows slowly. However, exponential functions eventually dominate both linear and quadratic functions. By $x = 10$ : $f(10) = 23$ , $g(10) = 99$ , $h(10) = 3072$ . The exponential function $h$ significantly outgrows both others despite starting slowly.
A researcher observes that when $x$ increases from 1 to 2, function $f$ increases by 5 units; when $x$ increases from 2 to 3, $f$ increases by 7 units; when $x$ increases from 3 to 4, $f$ increases by 9 units. Meanwhile, function $g$ doubles every time $x$ increases by 1. If both functions have the same value when $x = 1$ , which function will be larger when $x = 6$ ?
Answer: Function $g$ will be larger because exponential doubling eventually surpasses any polynomial growth pattern, regardless of initial conditions Function $f$ shows constant second differences (increases of 5, 7, 9 form first differences with second difference = 2), indicating quadratic growth. If $f(1) = k$ , then $f(6) = k + 5 + 7 + 9 + 11 + 13 = k + 45$ . Function $g$ doubles each step: $g(6) = g(1) \cdot 2^5 = k \cdot 32 = 32k$ . For any positive starting value $k$ , $32k > k + 45$ when $k > 1.45$ . Since both functions are increasing and start positive, exponential growth dominates.

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