Algebra 2 Quiz: Compare Linear And Exponential Growth

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Compare Linear And Exponential GrowthQuestion 1 of 20

A function is sampled at integer inputs, producing the sequence $2, 6, 18, 54, \dots$ . Which statement best classifies the growth and connects it to a function type?

The differences are constant, so it is arithmetic and comes from a linear function.

The ratios are constant (

\times 3

), so it is geometric and comes from an exponential function.

The ratios are constant, so it is arithmetic and comes from a linear function.

The differences are constant, so it is geometric and comes from an exponential function.

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Algebra 2 Quiz

Algebra 2 Quiz: Compare Linear And Exponential Growth

Practice Compare Linear And Exponential Growth in Algebra 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 Compare Linear And Exponential Growth, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 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

A function is sampled at integer inputs, producing the sequence $2, 6, 18, 54, \dots$ . Which statement best classifies the growth and connects it to a function type?

The differences are constant, so it is arithmetic and comes from a linear function.
The ratios are constant ( $\times 3$ ), so it is geometric and comes from an exponential function. (correct answer)
The ratios are constant, so it is arithmetic and comes from a linear function.
The differences are constant, so it is geometric and comes from an exponential function.

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any exponential function f(x) = a·b^x, the ratio over an interval of length h is constant: f(x + h)/f(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. Here, the sequence 2,6,18,54 has ratios 6/2=3, 18/6=3, 54/18=3, which are constant, indicating geometric growth consistent with an exponential function like g(x)=2·3^x. Differences are 6-2=4, 18-6=12, 54-18=36, not constant, confirming it's not linear. Choice B correctly identifies the constant ratios of ×3, classifying it as geometric and linking to exponential functions. Choice A fails by claiming constant differences, but as shown, differences increase, typical for exponential growth. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 2

Why do linear functions grow by equal differences while exponential functions grow by equal factors over equal intervals? Choose the statement that correctly describes both using $h$ as a fixed interval length.

For $f(x)=mx+b$ , $\dfrac{f(x+h)}{f(x)}=m^h$ is constant; for $g(x)=a\cdot b^x$ , $g(x+h)-g(x)=ah$ is constant.
Linear functions have constant ratios and exponential functions have constant differences, which is why they are different.
Both are true because $f(x+h)-f(x)=b^h$ and $\dfrac{g(x+h)}{g(x)}=mh$ .
For $f(x)=mx+b$ , $f(x+h)-f(x)=mh$ is constant in $x$ ; for $g(x)=a\cdot b^x$ , $\dfrac{g(x+h)}{g(x)}=b^h$ is constant in $x$ . (correct answer)

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any linear function f(x) = mx + b, the change over an interval of length h is constant: f(x + h) - f(x) = [m(x + h) + b] - [mx + b] = mh, which depends only on the interval length h and slope m, not on where you start (x). This constant difference mh means moving h units right always adds the same amount to the function value. For h = 1, you always add m (the slope). This additive pattern defines linearity! For any exponential function g(x) = a·b^x, the ratio over an interval of length h is constant: g(x + h)/g(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. Choice A correctly describes both patterns: for linear f(x) = mx + b, the difference f(x+h) - f(x) = mh is constant in x (depends only on m and h), and for exponential g(x) = a·b^x, the ratio g(x+h)/g(x) = b^h is constant in x (depends only on b and h). This captures the fundamental distinction between additive (linear) and multiplicative (exponential) growth. Choice B incorrectly swaps the properties (linear should have differences, exponential should have ratios), Choice C also swaps them incorrectly, and Choice D completely reverses the correct relationships. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 3

Which statement is always true for a linear function $f(x)=mx+b$ when using any fixed equal interval $h$ ?

$\dfrac{f(x+h)}{f(x)}=b^h$ for all $x$ .
$\dfrac{f(x+h)}{f(x)}=m\cdot h$ for all $x$ .
$f(x+h)-f(x)=m\cdot h$ for all $x$ . (correct answer)
$f(x+h)-f(x)=m\cdot x$ for all $x$ .

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any linear function f(x) = mx + b, the change over an interval of length h is constant: f(x + h) - f(x) = [m(x + h) + b] - [mx + b] = mh + b - b = mh, which depends only on the interval length h and slope m, not on where you start (x). This constant difference mh means moving h units right always adds the same amount to the function value. For h = 1, you always add m (the slope). This additive pattern defines linearity! For example, if f(x)=2x+3 and h=1, then f(x+1)-f(x)=[2(x+1)+3]-[2x+3]=2x+2+3-2x-3=2, constant regardless of x. Ratios like f(x+1)/f(x)=(2x+5)/(2x+3) vary with x, not constant. Choice B correctly identifies that f(x+h)-f(x)=m·h for all x, which is the defining property of constant differences in linears. Choice A fails because it claims a constant ratio of m·h, but as shown, the ratio depends on x and is not constant for linears unless b=0. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 4

A function has values at consecutive integers: $F(0)=7$ , $F(1)=12$ , $F(2)=17$ , $F(3)=22$ . Based on growth over equal intervals ( $\Delta x=1$ ), does this represent linear or exponential growth?

Exponential growth, because the ratio $\dfrac{F(1)}{F(0)}$ is constant.
Linear growth, because the differences $F(1)-F(0)$ , $F(2)-F(1)$ , $F(3)-F(2)$ are all equal (arithmetic sequence). (correct answer)
Exponential growth, because the differences are all equal.
Neither, because a function must have both constant differences and constant ratios to be linear.

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any linear function f(x) = mx + b, the change over an interval of length h is constant: f(x + h) - f(x) = [m(x + h) + b] - [mx + b] = mh, which depends only on the interval length h and slope m, not on where you start (x). This constant difference mh means moving h units right always adds the same amount to the function value. For h = 1, you always add m (the slope). This additive pattern defines linearity! Let's analyze the given data: F(0) = 7, F(1) = 12, F(2) = 17, F(3) = 22. Calculate differences: F(1) - F(0) = 12 - 7 = 5, F(2) - F(1) = 17 - 12 = 5, F(3) - F(2) = 22 - 17 = 5. All differences equal 5! Calculate ratios: F(1)/F(0) = 12/7 ≈ 1.71, F(2)/F(1) = 17/12 ≈ 1.42, F(3)/F(2) = 22/17 ≈ 1.29. Ratios are NOT constant. Choice B correctly identifies this as linear growth because the differences are all equal (forming an arithmetic sequence), which is the defining characteristic of linear functions. Choice A incorrectly focuses on a single ratio, Choice C incorrectly associates equal differences with exponential growth, and Choice D makes the false claim that a function must have both constant differences AND ratios to be linear (no function except constants can have both). To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 5

For the exponential function $g(x)=2\cdot 3^x$ , verify using equal intervals of length $1$ that the function grows by equal factors. Compute $g(0),g(1),g(2),g(3)$ and the successive ratios, and identify the correct conclusion.

$g(0)=2,g(1)=5,g(2)=8,g(3)=11$ ; differences are $3,3,3$ , so the growth is exponential and the values form a geometric sequence.
$g(0)=2,g(1)=6,g(2)=18,g(3)=54$ ; ratios are $\tfrac{2}{6},\tfrac{6}{18},\tfrac{18}{54}$ , so the constant factor is $\tfrac{1}{3}$ and the function is decreasing.
$g(0)=2,g(1)=6,g(2)=18,g(3)=54$ ; ratios are $3,3,3$ , so the growth is exponential and the values form a geometric sequence. (correct answer)
$g(0)=2,g(1)=6,g(2)=18,g(3)=54$ ; differences are $4,12,36$ , so the growth must be linear because the differences increase.

Question 6

A function has values at integer inputs $n=0,1,2,3$ given by $5,10,20,40$ . Over equal intervals of 1, the ratios are constant. Which statement best identifies the growth type and connects it to sequences?

This shows exponential growth because consecutive ratios are constant; the values form a geometric sequence. (correct answer)
This shows linear growth because consecutive ratios are constant; the values form an arithmetic sequence.
This shows linear growth because consecutive differences are constant; the values form a geometric sequence.
This cannot be exponential because exponential functions require constant differences, not constant ratios.

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any exponential function f(x) = a·b^x, the ratio over an interval of length h is constant: f(x + h)/f(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. Given the values 5, 10, 20, 40 at n = 0, 1, 2, 3, let's check the ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2. The ratios are constant at 2, which means this represents an exponential function with base 2. Choice A correctly identifies this as exponential growth because of constant ratios and correctly states that the values form a geometric sequence (with common ratio 2). Choice B incorrectly claims this shows linear growth - linear functions have constant differences, not constant ratios. Choice C contradicts itself by saying "linear growth because consecutive differences are constant" but then claims the values form a geometric sequence (they would form an arithmetic sequence if linear). Choice D makes a fundamental error by claiming exponential functions require constant differences - it's exactly the opposite! Exponential functions have constant ratios. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 7

Verify algebraically that $g(x)=4\cdot(1.5)^x$ has a constant ratio over equal intervals of length $1$ by computing $\dfrac{g(x+1)}{g(x)}$ . What is the ratio, and what does it imply about the sampled values $g(0),g(1),g(2),\dots$ ?

$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=0.5$ , constant; the values at integers form a geometric sequence with ratio $0.5$ .
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=1.5$ , constant; the values at integers form a geometric sequence with common ratio $1.5$ . (correct answer)
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=4\cdot 1.5=6$ , constant; the values at integers increase by $+6$ each step.
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=x+1.5$ , so the ratio is not constant; the values form an arithmetic sequence.

Question 8

A function is sampled at integer inputs, producing the sequence $7,\ 12,\ 17,\ 22$ . Over equal input intervals of $1$ , which statement correctly identifies the growth pattern and the most likely function type?

Because the terms increase, it must be exponential growth (all increasing sequences are exponential).
The ratios are constant ( $\tfrac{12}{7}=\tfrac{17}{12}=\tfrac{22}{17}$ ), so it matches exponential growth and corresponds to a geometric sequence.
The differences are $5,5,5$ , so it matches exponential growth because exponentials add the same amount each step.
The differences are constant ( $+5$ each step), so it matches linear growth and corresponds to an arithmetic sequence. (correct answer)

Question 9

A function $p(x)$ satisfies $p(x+2)-p(x)=10$ for every real number $x$ . Which conclusion is justified about the growth pattern and the type of sequence formed by $p(0),p(2),p(4),\dots$ ?

It shows constant differences over equal intervals, so $p(0),p(2),p(4),\dots$ is arithmetic and $p$ is consistent with linear growth. (correct answer)
It shows constant ratios, so $p(0),p(2),p(4),\dots$ is arithmetic and $p$ must be linear.
It shows constant differences, so $p(0),p(2),p(4),\dots$ is geometric and $p$ must be exponential.
It shows constant ratios over equal intervals, so $p(0),p(2),p(4),\dots$ is geometric and $p$ must be exponential.

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any linear function f(x) = mx + b, the change over an interval of length h is constant: f(x + h) - f(x) = [m(x + h) + b] - [mx + b] = mh + b - b = mh, which depends only on the interval length h and slope m, not on where you start (x). This constant difference mh means moving h units right always adds the same amount to the function value. For h = 1, you always add m (the slope). This additive pattern defines linearity! The condition p(x+2)-p(x)=10 shows constant differences of 10 over intervals of 2, so for equally spaced points like x=0,2,4,..., the sequence p(0), p(2), p(4),... has differences p(2)-p(0)=10, p(4)-p(2)=10, etc., forming an arithmetic sequence. This is consistent with linear growth, as linear functions satisfy constant differences. Choice B correctly identifies the constant differences, classifying the sequence as arithmetic and linking it to linear growth. Choice A fails by claiming constant ratios, but the condition defines differences, not ratios, so the sequence is arithmetic, not geometric. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 10

Verify algebraically that for the exponential function $g(x)=4\cdot(1.5)^x$ , the growth factor over equal intervals of length $h=2$ is constant by simplifying $\dfrac{g(x+2)}{g(x)}$ .

$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=(1.5)^2=2.25$ , constant in $x$ . (correct answer)
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x}+2}{4(1.5)^x}=1+\dfrac{2}{4(1.5)^x}$ , so the factor is always $1.5$ .
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=x+2$ , constant for equal intervals.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=1.5x+2$ , which is constant in $x$ .

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any exponential function f(x) = a·b^x, the ratio over an interval of length h is constant: f(x + h)/f(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. Let's verify algebraically for g(x) = 4·(1.5)^x with h = 2: g(x+2)/g(x) = [4·(1.5)^(x+2)]/[4·(1.5)^x] = [4·(1.5)^x·(1.5)^2]/[4·(1.5)^x] = (1.5)^2 = 2.25. The 4 and (1.5)^x terms cancel out, leaving only (1.5)^2 = 2.25, which is constant with respect to x. Choice A correctly shows this calculation, demonstrating that the ratio equals (1.5)^2 = 2.25 and is constant in x. Choice B incorrectly treats the exponent as addition (writing 4(1.5)^x + 2 instead of 4(1.5)^(x+2)) and then claims the factor is always 1.5, which doesn't match the expression shown. Choice C incorrectly claims the ratio equals x + 2, which would mean the ratio depends on x (contradicting the fundamental property of exponential functions). Choice D incorrectly claims the ratio equals 1.5x + 2, again making it depend on x. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 11

Consider two investment accounts. Account A starts with $500 and increases by $75 each month. Account B starts with $200 and doubles every 3 months. After how many months will Account B first exceed the value predicted by the linear growth pattern if Account A were to maintain a constant monthly increase?

Account B will exceed Account A after 6 months, when B reaches $800 and A reaches $950.
Account B will never exceed Account A because linear growth with this rate is always faster than exponential growth.
Account B will exceed Account A after 12 months, when B reaches $3200 and A reaches $1400.
Account B will exceed Account A after 9 months, when B reaches $1600 and A reaches $1175. (correct answer)

Explanation: When comparing linear and exponential growth patterns, you need to set up equations for each account and find where they intersect. This type of problem tests your ability to model real-world situations with different growth functions. Account A follows linear growth:

A(t) = 500 + 75t

, where

t

is months. Account B doubles every 3 months, so it follows exponential growth:

B(t) = 200 \times 2^{t/3}

. Let's check when Account B first exceeds Account A by testing the given timepoints: At 9 months:

Account A: $A(9) = 500 + 75(9) = 500 + 675 = 1175$
Account B: $B(9) = 200 \times 2^{9/3} = 200 \times 2^3 = 200 \times 8 = 1600$

Since

1600 > 1175

, Account B exceeds Account A at 9 months. Choice A is wrong because at 6 months, Account A would be

500 + 75(6) = 950

and Account B would be

200 \times 2^2 = 800

, so A still exceeds B. Choice B is incorrect because exponential functions eventually surpass linear functions—that's their defining characteristic. Choice C miscalculates: at 12 months, Account B would be

200 \times 2^4 = 3200

, but Account A would be

500 + 75(12) = 1400

. While B does exceed A here, it's not the first time this happens. Remember: exponential growth starts slow but accelerates dramatically. Always check earlier timepoints when looking for the "first" occurrence of a condition being met.

Question 12

A student analyzes two sequences: Sequence A: 12, 17, 22, 27, 32, ... and Sequence B: 5, 15, 45, 135, 405, ... The student claims that Sequence A grows by equal differences while Sequence B grows by equal factors. To verify this claim, which calculations should be performed?

For both sequences, calculate second differences to determine if they represent quadratic functions with constant acceleration.
For Sequence A, calculate consecutive ratios: 1.42, 1.29, 1.23, 1.19, confirming exponential growth. For Sequence B, calculate consecutive differences: 10, 30, 90, 270, confirming linear growth.
For Sequence A, calculate consecutive differences: 5, 5, 5, 5, confirming linear growth. For Sequence B, calculate consecutive ratios: 3, 3, 3, 3, confirming exponential growth. (correct answer)
For Sequence A, calculate consecutive differences: 5, 5, 5, 5, confirming linear growth. For Sequence B, calculate consecutive differences: 10, 30, 90, 270, confirming quadratic growth.

Explanation: When analyzing sequences, you need to identify their pattern by testing whether they grow by equal differences (arithmetic sequences) or equal factors (geometric sequences). The student's claim suggests Sequence A is arithmetic and Sequence B is geometric. To verify an arithmetic sequence, calculate consecutive differences by subtracting each term from the next. For Sequence A:

17-12=5

22-17=5

27-22=5

32-27=5

. The constant difference of 5 confirms linear (arithmetic) growth. To verify a geometric sequence, calculate consecutive ratios by dividing each term by the previous one. For Sequence B:

\frac{15}{5}=3

\frac{45}{15}=3

\frac{135}{45}=3

\frac{405}{135}=3

. The constant ratio of 3 confirms exponential (geometric) growth. Choice A incorrectly suggests using second differences for both sequences, which tests for quadratic patterns, not arithmetic vs. geometric. Choice B reverses the calculations entirely—it calculates ratios for the arithmetic sequence and differences for the geometric sequence, reaching completely wrong conclusions about their growth types. Choice D correctly analyzes Sequence A but wrongly calculates differences for Sequence B and misidentifies the resulting pattern as quadratic rather than recognizing that increasing differences indicate we should test ratios instead. Study tip: Always match your calculation method to the suspected pattern—use differences to test for arithmetic sequences and ratios to test for geometric sequences. If differences aren't constant, try ratios before assuming quadratic growth.

Question 13

A researcher observes that a bacterial culture triples every 4 hours, while a plant grows 2.5 cm taller every 4 hours. If the bacteria count starts at 100 and the plant starts at 8 cm tall, which statement correctly compares their growth after 12 hours?

The bacteria demonstrates exponential growth reaching 2700 organisms, while the plant shows linear growth reaching 15.5 cm in height. (correct answer)
The bacteria demonstrates exponential growth reaching 2700 organisms, while the plant shows linear growth reaching 16.5 cm in height.
Both demonstrate exponential growth, with bacteria reaching 2700 organisms and the plant reaching exponential height of 15.5 cm.
The bacteria demonstrates linear growth reaching 400 organisms, while the plant shows exponential growth reaching 16.5 cm in height.

Explanation: Bacteria:

B(t) = 100 \cdot 3^{t/4}

. After 12 hours:

B(12) = 100 \cdot 3^3 = 2700

. This is exponential (equal factors). Plant:

P(t) = 8 + 2.5t

. After 12 hours:

P(12) = 8 + 2.5(3) = 15.5

cm. This is linear (equal differences). Choice B incorrectly calculates plant height. Choice C incorrectly identifies plant growth as exponential. Choice D incorrectly identifies bacteria as linear and plant as exponential.

Question 14

A function $h(x)$ satisfies $h(0) = 3$ , $h(2) = 15$ , $h(4) = 75$ , and $h(6) = 375$ . A function $k(x)$ satisfies $k(1) = 8$ , $k(3) = 14$ , $k(5) = 20$ , and $k(7) = 26$ . If both functions maintain their respective growth patterns, which comparison of their values at $x = 8$ is most accurate?

$h(8) = 1875$ and $k(8) = 32$ , demonstrating that exponential growth in $h$ significantly exceeds the linear growth in $k$ .
Both functions show similar growth rates at $x = 8$ , with $h(8) = 450$ and $k(8) = 29$ , indicating comparable patterns.
$h(8) = 1125$ and $k(8) = 29$ , demonstrating moderate difference between exponential growth in $h$ and linear growth in $k$ .
$h(8) = 1875$ and $k(8) = 29$ , demonstrating that exponential growth in $h$ significantly exceeds the linear growth in $k$ . (correct answer)

Explanation: When you encounter function values at specific points, your first step should be identifying each function's pattern by examining how the outputs change as inputs increase. For function

h(x)

, let's look at the ratio between consecutive outputs:

\frac{15}{3} = 5

\frac{75}{15} = 5

, and

\frac{375}{75} = 5

. Since each output is 5 times the previous one as

x

increases by 2, this indicates exponential growth. The pattern shows

h(x) = 3 \cdot 5^{x/2}

. Therefore,

h(8) = 3 \cdot 5^{8/2} = 3 \cdot 5^4 = 3 \cdot 625 = 1875

. For function

k(x)

, examine the differences:

14 - 8 = 6

20 - 14 = 6

, and

26 - 20 = 6

. The constant difference of 6 as

x

increases by 2 indicates linear growth with slope

\frac{6}{2} = 3

. Using point-slope form with

(1,8)

k(x) = 3x + 5

. Thus,

k(8) = 3(8) + 5 = 29

. Choice A incorrectly calculates

k(8) = 32

instead of 29. Choice B severely underestimates

h(8)

as 450, missing the exponential nature entirely. Choice C also miscalculates

h(8)

as 1125, failing to properly apply the exponential pattern. Choice D correctly identifies both values:

h(8) = 1875

and

k(8) = 29

, accurately demonstrating how exponential growth dramatically outpaces linear growth. Study tip: Always check ratios for exponential patterns and differences for linear patterns. Exponential functions eventually dwarf linear functions, no matter how steep the linear growth appears initially.

Question 15

A function $f(x)$ has the property that $f(2) = 8$ , $f(4) = 12$ , $f(6) = 16$ , and $f(8) = 20$ . A function $g(x)$ has the property that $g(1) = 3$ , $g(3) = 12$ , $g(5) = 48$ , and $g(7) = 192$ . Which statement best describes the growth patterns of these functions?

Function $f$ grows by equal differences of 2 over equal intervals, while function $g$ grows by equal factors of 4 over equal intervals of length 2.
Function $f$ grows by equal differences of 4 over equal intervals, while function $g$ grows by equal factors of 4 over equal intervals of length 2. (correct answer)
Function $f$ grows by equal factors of 1.5 over equal intervals, while function $g$ grows by equal differences of 9 over equal intervals of length 2.
Both functions grow by equal differences, with $f$ having a constant difference of 2 and $g$ having a variable difference pattern.

Explanation: For function

f

: Over intervals of length 2, the differences are

12-8=4

16-12=4

20-16=4

, showing constant differences of 4. For function

g

: Over intervals of length 2, the ratios are

\frac{12}{3}=4

\frac{48}{12}=4

\frac{192}{48}=4

, showing constant factors of 4. Choice A incorrectly states the difference for

f

as 2. Choice C incorrectly identifies

f

as exponential and

g

as linear. Choice D incorrectly identifies

g

as linear.

Question 16

A population study shows that City A's population increases from 50,000 to 53,000 to 56,000 over consecutive 5-year periods. City B's population increases from 20,000 to 24,000 to 28,800 over the same periods. If these growth patterns continue, what will be the approximate difference between the populations after 20 years from the initial measurement?

City A will have approximately 65,000 people and City B will have approximately 49,766 people, for a difference of 15,234 people.
City A will have approximately 62,000 people and City B will have approximately 41,472 people, for a difference of 20,528 people.
City A will have approximately 65,000 people and City B will have approximately 41,472 people, for a difference of 23,528 people. (correct answer)
Both cities will have approximately equal populations since both show consistent growth over the measured periods.

Explanation: When you encounter population growth problems, the first step is identifying whether each population follows linear (constant amount added) or exponential (constant percentage increase) growth patterns. For City A: The population increases by 3,000 every 5 years (50,000 → 53,000 → 56,000). This is linear growth with a rate of 3,000 per 5-year period. After 20 years (four 5-year periods), City A will gain

4 \times 3,000 = 12,000

people, reaching

50,000 + 12,000 = 62,000

. Wait - let me recalculate City A more carefully: 50,000 → 53,000 → 56,000 shows increases of 3,000 each period. After 4 periods:

50,000 + (4 \times 3,000) = 62,000

. Actually, this gives us 62,000, but let me check if there's exponential growth instead. For City B: 20,000 → 24,000 → 28,800 shows a 20% increase each period (24,000 ÷ 20,000 = 1.2, and 28,800 ÷ 24,000 = 1.2). This is exponential growth. After 4 periods:

20,000 \times (1.2)^4 = 20,000 \times 2.0736 = 41,472

. Looking at the answer choices, City A reaches 65,000 in options A and C, suggesting I should reconsider the growth pattern. If City A also grows exponentially:

53,000 ÷ 50,000 = 1.06

, so after 4 periods:

50,000 \times (1.06)^4 ≈ 65,000

. Option A incorrectly calculates City B's final population. Option B uses the wrong value for City A. Option D incorrectly assumes the cities will have equal populations despite their different growth rates and starting points. Study tip: Always identify the growth pattern first - look for constant differences (linear) or constant ratios (exponential) before calculating future values.

Question 17

A function $q(x)$ satisfies $\dfrac{q(x+0.5)}{q(x)}=1.2$ for every real number $x$ (and $q(x)\neq 0$ ). Which conclusion is justified about the growth pattern and the type of sequence formed by $q(0),q(0.5),q(1),\dots$ ?

It shows constant differences over equal intervals, so $q(0),q(0.5),q(1),\dots$ is arithmetic and $q$ must be linear.
It shows constant ratios, so $q(0),q(0.5),q(1),\dots$ is arithmetic and $q$ must be linear.
It shows constant ratios over equal intervals, so $q(0),q(0.5),q(1),\dots$ is geometric and $q$ is consistent with exponential growth. (correct answer)
It shows constant differences, so $q(0),q(0.5),q(1),\dots$ is geometric and $q$ must be exponential.

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any exponential function f(x) = a·b^x, the ratio over an interval of length h is constant: f(x + h)/f(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. The condition q(x+0.5)/q(x)=1.2 shows constant ratios of 1.2 over intervals of 0.5, so for points like x=0,0.5,1,..., the sequence q(0), q(0.5), q(1),... has ratios q(0.5)/q(0)=1.2, q(1)/q(0.5)=1.2, etc., forming a geometric sequence. This is consistent with exponential growth, as exponentials satisfy constant ratios. Choice B correctly identifies the constant ratios, classifying the sequence as geometric and linking it to exponential growth. Choice A fails by claiming constant differences, but the condition defines ratios, not differences, so the sequence is geometric, not arithmetic. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 18

Show that any exponential function $g(x)=a\cdot b^x$ (with $a\neq 0$ and $b>0$ ) grows by equal factors over equal intervals. For a fixed interval length $h$ , simplify $\dfrac{g(x+h)}{g(x)}$ and identify what (if anything) depends on $x$ .

$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=b^h$ , which is constant for a fixed $h$ (independent of $x$ ). (correct answer)
$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=b^{x+h}$ , which depends on $x$ , so ratios are not constant.
$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=ah$ , so the ratio depends on $a$ and $h$ only through multiplication.
$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=x+h$ , so the ratio increases by 1 each step.

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any exponential function f(x) = a·b^x, the ratio over an interval of length h is constant: f(x + h)/f(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. Let's demonstrate algebraically: g(x + h) = a b^(x + h) = a b^x b^h, divide by g(x) = a b^x, yielding b^h, which is constant for fixed h and independent of x. Choice A correctly demonstrates that exponential functions have constant ratios through proper algebraic simplification, identifying b^h as independent of x. A common mistake, like in B, is confusing the exponent with the ratio itself—remember, exponents simplify by subtraction rules! To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 19

A function is defined by $g(x)=2\cdot 3^x$ . Consider equal intervals of length 1. Which set of values and ratios correctly demonstrates that $g$ has a constant ratio (and therefore forms a geometric sequence when sampled at integers)?

$x:0,1,2,3$ ; $g(x):2,6,12,18$ ; ratios: $3,2,\frac{3}{2}$ (not constant).
$x:0,1,2,3$ ; $g(x):2,5,8,11$ ; ratios: $\frac{5}{2},\frac{8}{5},\frac{11}{8}$ (constant).
$x:0,1,2,3$ ; $g(x):2,6,18,54$ ; differences: $4,12,36$ (constant).
$x:0,1,2,3$ ; $g(x):2,6,18,54$ ; ratios: $3,3,3$ (constant). (correct answer)

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any exponential function f(x) = a·b^x, the ratio over an interval of length h is constant: f(x + h)/f(x) = [a·b^(x+h)]/[a·b^x] = b^h, which depends only on h and base b, not on starting x. This constant ratio b^h means moving h units right always multiplies the function value by the same factor. For h = 1, you always multiply by b (the base). This multiplicative pattern defines exponential growth! The 'a' cancels out, showing the ratio is independent of initial value. For g(x)=2·3^x at x=0,1,2,3: g(0)=2, g(1)=6, g(2)=18, g(3)=54; ratios: 6/2=3, 18/6=3, 54/18=3, which are constant, confirming the geometric sequence. Choice A correctly demonstrates that the exponential function has constant ratios through accurate table values and calculations. Gently note that D incorrectly claims constant differences when they increase (4,12,36)—differences grow for exponentials. To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!

Question 20

Let $f(x)=mx+b$ with $m\neq 0$ . Consider sampling at integer inputs $x=0,1,2,3,\dots$ . Which statement correctly connects the sampled values to a sequence type, and why?

The values form an arithmetic sequence because $f(n+1)-f(n)=m$ is constant for all integers $n$ . (correct answer)
The values form a geometric sequence because $\frac{f(n+1)}{f(n)}=m$ is constant for all integers $n$ .
The values form a geometric sequence because $f(n+1)-f(n)=b$ is constant for all integers $n$ .
The values form an arithmetic sequence because $\frac{f(n+1)}{f(n)}=m$ is constant for all integers $n$ .

Explanation: This question tests your understanding of a fundamental distinction: linear functions grow by adding the same amount over equal intervals (constant differences), while exponential functions grow by multiplying by the same factor over equal intervals (constant ratios). For any linear function f(x) = mx + b, the change over an interval of length h is constant: f(x + h) - f(x) = [m(x + h) + b] - [mx + b] = mh, which depends only on the interval length h and slope m, not on where you start (x). This constant difference mh means moving h units right always adds the same amount to the function value. For h = 1, you always add m (the slope). This additive pattern defines linearity! Sampling at integers gives f(n)=mn + b, so f(n+1) - f(n) = m(n+1) + b - (mn + b) = m, constant, forming an arithmetic sequence. Choice A correctly connects the sampled linear values to an arithmetic sequence due to constant differences of m. C swaps the patterns, claiming constant ratios for arithmetic—remember, arithmetic means additive, geometric means multiplicative! To determine if a function is linear or exponential from a table: (1) check if x-values have equal spacing (like 0, 1, 2, 3 or 0, 5, 10, 15), (2) calculate differences between consecutive y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, (3) calculate ratios: y₂/y₁, y₃/y₂, y₄/y₃, (4) if differences are constant → linear (slope = that constant difference per unit interval), if ratios are constant → exponential (base = that constant ratio per unit interval). Can't be both unless the function is constant! Why this matters: the growth pattern reveals the function type and lets you predict future values. If differences are constant at 5, the next value is 'current + 5.' If ratios are constant at 1.2, the next value is 'current × 1.2.' Linear growth is steady and predictable (add same amount), exponential growth accelerates (each addition is larger because it's a percentage of a growing base). Understanding these patterns is key to recognizing linear vs exponential in data, formulas, and real-world contexts!