Compare Linear and Exponential Growth
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Algebra 2 › Compare Linear and Exponential Growth
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 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 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!
A function is sampled at integer inputs, producing the sequence $5, 8, 11, 14, \dots$. Which statement best classifies the growth and connects it to a function type?
The differences are constant ($+3$), so it is arithmetic and comes from a linear function.
Neither differences nor ratios are constant, so it cannot come from any function.
The ratios are constant, so it is geometric and comes from an exponential 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 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! Here, the sequence 5,8,11,14 has differences 8-5=3, 11-8=3, 14-11=3, which are constant, indicating arithmetic growth consistent with a linear function like f(x)=3x+5 (for x=0,1,2,3,...). Ratios are 8/5=1.6, 11/8≈1.375, 14/11≈1.273, not constant, confirming it's not exponential. Choice B correctly identifies the constant differences of +3, classifying it as arithmetic and linking to linear functions. Choice A fails by claiming constant ratios, but as calculated, the ratios vary, so it's not geometric or exponential. 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!
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.
Linear functions have constant ratios and exponential functions have constant differences, which is why they are different.
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.
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 $x$.
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!
Which statement is always true for a linear function $f(x)=mx+b$ when using any fixed equal interval $h$?
$f(x+h)-f(x)=m\cdot x$ for all $x$.
$f(x+h)-f(x)=m\cdot h$ for all $x$.
$\dfrac{f(x+h)}{f(x)}=m\cdot h$ for all $x$.
$\dfrac{f(x+h)}{f(x)}=b^h$ 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!
Verify constant-ratio growth over an equal interval of length $h=2$ for the exponential function $g(x)=4\cdot(1.5)^x$ by simplifying $\dfrac{g(x+2)}{g(x)}$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=1.5^x$, so the ratio depends on $x$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=1.5x+2$, so the ratio depends on $x$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=(1.5)^2=2.25$, a constant independent of $x$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=\dfrac{2}{1.5}=\frac{4}{3}$, a constant factor.
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 h=2: g(x+2)/g(x) = [4 $(1.5)^{x+2}$] / [4 $(1.5)^x$] = $(1.5)^{x+2}$ / $(1.5)^x$ = $(1.5)^2$ = 2.25, constant and independent of x. Choice A correctly proves constant ratios for the exponential through proper algebra, yielding 2.25 independent of x. D miscalculates the exponent $simplification—(1.5)^{x+2}$ / $(1.5)^x$ = $(1.5)^2$, not 2/1.5! 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!
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?
Linear growth, because the differences $F(1)-F(0)$, $F(2)-F(1)$, $F(3)-F(2)$ are all equal (arithmetic sequence).
Exponential growth, because the differences are all equal.
Exponential growth, because the ratio $\dfrac{F(1)}{F(0)}$ is constant.
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!
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.
$g(0)=2,g(1)=6,g(2)=18,g(3)=54$; 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.
$g(0)=2,g(1)=6,g(2)=18,g(3)=54$; differences are $4,12,36$, so the growth must be linear because the differences increase.
$g(0)=2,g(1)=5,g(2)=8,g(3)=11$; differences are $3,3,3$, so the growth is exponential and the values form a geometric sequence.
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 the exponential function g(x) = $2·3^x$, let's calculate: g(0) = $2·3^0$ = 2·1 = 2, g(1) = $2·3^1$ = 2·3 = 6, g(2) = $2·3^2$ = 2·9 = 18, g(3) = $2·3^3$ = 2·27 = 54. The successive ratios are: g(1)/g(0) = 6/2 = 3, g(2)/g(1) = 18/6 = 3, g(3)/g(2) = 54/18 = 3. All ratios equal 3, confirming constant ratios! Choice A correctly identifies the values as 2, 6, 18, 54 with constant ratios of 3, proving exponential growth and forming a geometric sequence with common ratio 3 (which equals the base b = 3). Choice B shows values 2, 5, 8, 11 which would be linear, not exponential. Choice C correctly calculates the values but focuses on differences (4, 12, 36) which increase - this is expected for exponentials but doesn't define them. Choice D makes a calculation error by inverting the ratios (should be 6/2, not 2/6) and incorrectly concludes the function is decreasing. To determine if a function is linear or exponential from a table: calculate ratios between consecutive y-values - if they're constant, the function is exponential and the constant ratio equals the base. Why this matters: recognizing constant ratios immediately tells you the function type and lets you predict future values by multiplying by that constant!
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 linear growth because consecutive ratios are constant; the values form an arithmetic sequence.
This shows exponential growth because consecutive ratios are constant; the values form a geometric sequence.
This cannot be exponential because exponential functions require constant differences, not constant ratios.
This shows linear growth because consecutive differences are constant; the values form a geometric sequence.
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!
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 $g(0),g(1),g(2),\dots$?
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=1.5$, constant; the values at integers form a geometric sequence with common ratio $1.5$.
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=4\cdot 1.5=6$, constant; the values at integers increase by $+6$ each step.
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=0.5$, constant; the values at integers form a geometric sequence with ratio $0.5$.
$\dfrac{g(x+1)}{g(x)}=\dfrac{4(1.5)^{x+1}}{4(1.5)^x}=x+1.5$, so the ratio is not constant; the values form an arithmetic sequence.
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 g(x) = $4·(1.5)^x$, let's compute g(x+1)/g(x): g(x+1)/g(x) = [4·(1.5)^(x+1)] / $[4·(1.5)^x$] = $[4·(1.5)^x$$·(1.5)^1$] / $[4·(1.5)^x$] = $(1.5)^1$ = 1.5. The 4's cancel and we use the exponent rule b^(x+1) = $b^x$$·b^1$. The ratio is constant at 1.5! Choice A correctly shows the algebraic simplification yielding a constant ratio of 1.5, proving that g(0), g(1), g(2),... forms a geometric sequence with common ratio 1.5. Choice B incorrectly claims the ratio equals x + 1.5, suggesting it depends on x. Choice C incorrectly calculates the ratio as 0.5 instead of 1.5. Choice D incorrectly multiplies 4·1.5 = 6 and confuses ratio with difference, claiming values increase by +6. To verify: g(0) = 4, g(1) = 6, g(2) = 9, g(3) = 13.5. Check ratios: 6/4 = 1.5, 9/6 = 1.5, 13.5/9 = 1.5. Constant! Why this matters: the constant ratio 1.5 means each value is 1.5 times the previous one, defining the geometric sequence and confirming exponential growth with base 1.5!
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?
The differences are constant ($+5$ each step), so it matches linear growth and corresponds to an arithmetic sequence.
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.
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). Given the sequence 7, 12, 17, 22, let's check both patterns. Differences: 12 - 7 = 5, 17 - 12 = 5, 22 - 17 = 5. All differences equal 5 - constant! Ratios: 12/7 ≈ 1.71, 17/12 ≈ 1.42, 22/17 ≈ 1.29. The ratios are different - not constant! Choice A correctly identifies constant differences of +5, indicating linear growth and an arithmetic sequence. This suggests the function is f(x) = 5x + 7. Choice B incorrectly claims the ratios are constant (they're not equal: 12/7 ≠ 17/12 ≠ 22/17). Choice C makes the false claim that all increasing sequences are exponential - many increasing sequences are linear! Choice D correctly identifies the differences but wrongly associates constant differences with exponential growth instead of linear growth. To determine if a function is linear or exponential from a table: (1) calculate differences between consecutive y-values, (2) calculate ratios between consecutive y-values, (3) if differences are constant → linear, if ratios are constant → exponential. Why this matters: the growth pattern reveals the function type and lets you predict the next value - here, add 5 to get 27!