Math 2 Quiz: Comparing Function Types

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Comparing Function TypesQuestion 1 of 6

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?

The balloon will eventually be highest because exponential functions with base greater than 1 increase without bound over time.

The balloon will be highest initially but the ball will surpass it during the ball's upward trajectory phase.

The balloon will be highest only between t = 2 and t = 3 seconds when its exponential growth temporarily exceeds the others.

The balloon will never be highest because both the quadratic ball function and linear elevator function start with advantages.

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Math 2 Quiz

Math 2 Quiz: Comparing Function Types

Practice Comparing Function Types in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Comparing Function Types, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The balloon will eventually be highest because exponential functions with base greater than 1 increase without bound over time. (correct answer)
The balloon will be highest initially but the ball will surpass it during the ball's upward trajectory phase.
The balloon will be highest only between t = 2 and t = 3 seconds when its exponential growth temporarily exceeds the others.
The balloon will never be highest because both the quadratic ball function and linear elevator function start with advantages.

Explanation: 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.

Question 2

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?

City 1 because linear growth with 1200 people per year will accumulate to the largest total over three decades.
City 2 because exponential growth at 3% annually compounds to exceed both linear and quadratic growth over 30 years. (correct answer)
City 3 because quadratic growth accelerates faster than linear and will surpass the exponential model's 3% rate.
City 2 because it starts with the largest initial population and exponential growth preserves this advantage.

Explanation: 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.

Question 3

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?

Species A will reach the limit first at approximately $t = 26.7$ because linear growth is most predictable and consistent
Species B will reach the limit first at approximately $t = 20.5$ because quadratic growth accelerates faster than linear growth initially
Species C will reach the limit first at approximately $t = 12.8$ because exponential growth eventually surpasses both linear and quadratic growth (correct answer)
All three species will reach the carrying capacity at nearly the same time since they start with similar initial populations

Explanation: 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.

Question 4

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?

Function $p(x)$ will dominate because quadratic functions with positive leading coefficients grow without bound more predictably than exponential functions
Function $q(x)$ will dominate because exponential functions with base $r > 1$ eventually grow faster than any polynomial function, regardless of coefficients (correct answer)
The dominant function depends on the specific values of the parameters, making it impossible to determine without solving for $a, b, c, d,$ and $r$
Both functions will grow at comparable rates since they pass through the same two points, indicating similar fundamental growth patterns

Explanation: 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.

Question 5

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?

Function $f$ grows fastest initially, but $h$ eventually surpasses both $f$ and $g$ in growth rate
Function $g$ maintains the highest growth rate throughout the entire interval since quadratic functions grow faster than linear functions
Function $h$ grows slowest initially, but its exponential nature causes it to eventually grow faster than both linear and quadratic functions (correct answer)
All three functions have comparable growth rates over this interval, with no single function significantly outpacing the others

Explanation: 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.

Question 6

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 $x = 6$ ?

Function $f$ will be larger because its consistent acceleration in growth rate provides steady long-term gains over exponential growth
Function $g$ will be larger because exponential doubling eventually surpasses any polynomial growth pattern, regardless of initial conditions (correct answer)
Both functions will have approximately the same value since they start equal and show comparable growth patterns over this interval
The comparison cannot be determined without knowing the specific equations or additional data points for both functions

Explanation: 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.