Compare Linear and Exponential Growth
Help Questions
Algebra 2 › Compare Linear and Exponential Growth
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, so it is geometric and comes from an exponential function.
The ratios are constant, so it is geometric and comes from an exponential function.
Neither differences nor ratios are constant, so it cannot come from any function.
The differences are constant ($+3$), so it is arithmetic and comes from a linear 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!
Let $g(x)=a\cdot b^x$ with $a\neq 0$ and $b\neq 1$. 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 a geometric sequence because $\frac{g(n+1)}{g(n)}=b$ is constant for all integers $n$.
The values form an arithmetic sequence because $\frac{g(n+1)}{g(n)}=n+1$ is constant for all integers $n$.
The values form a geometric sequence because $g(n+1)-g(n)=a$ is constant for all integers $n$.
The values form an arithmetic sequence because $g(n+1)-g(n)=b$ 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 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. Sampling at integers gives g(n)=a $b^n$, so g(n+1)/g(n) = (a $b^{n+1}$) / (a $b^n$) = b, constant, forming a geometric sequence. Choice A correctly connects the sampled exponential values to a geometric sequence due to constant ratios of b. B confuses it with arithmetic by claiming constant differences of b—exponentials have growing differences! 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 correctly contrasts linear vs. exponential growth over equal input intervals of length $h$ using differences and ratios?
Exponential functions have constant differences because $g(x+h)-g(x)=ah$, and linear functions have constant ratios because $\frac{f(x+h)}{f(x)}=b^h$.
Linear functions have constant ratios because $\frac{f(x+h)}{f(x)}=m$, while exponential functions have constant differences because $g(x+h)-g(x)=b$.
Linear $f(x)=mx+b$ has constant differences because $f(x+h)-f(x)=mh$, while exponential $g(x)=a\cdot b^x$ has constant ratios because $\frac{g(x+h)}{g(x)}=b^h$.
Both linear and exponential functions have constant differences and constant ratios over equal intervals.
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 exponential g(x)=a $b^x$, $g(x+h)/g(x)=b^h$, a constant ratio, highlighting the multiplicative nature. Choice A correctly contrasts linear constant differences (mh) with exponential constant ratios $(b^h$) using accurate formulas. B swaps the patterns, assigning ratios to linear and differences to exponential—keep the additive vs. multiplicative distinction clear! 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!
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 $x$.
$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=ah$, 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.
$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=b^{x+h}$, which depends on $x$, so ratios are not constant.
$\dfrac{g(x+h)}{g(x)}=\dfrac{a b^{x+h}}{a b^x}=b^h$, which is constant for a fixed $h$ (independent of $x$).
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 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!
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)?
$x:0,1,2,3$; $g(x):2,6,18,54$; ratios: $3,3,3$ (constant).
$x:0,1,2,3$; $g(x):2,6,12,18$; 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).
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!
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 $\dfrac{g(x+2)}{g(x)}$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=(1.5)^2=2.25$, a constant independent of $x$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=1.5^x$, so the ratio depends on $x$.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=\dfrac{2}{1.5}=\frac{4}{3}$, a constant factor.
$\dfrac{g(x+2)}{g(x)}=\dfrac{4(1.5)^{x+2}}{4(1.5)^x}=1.5x+2$, so the ratio depends on $x$.
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!
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 $\frac{f(n+1)}{f(n)}=m$ is constant for all integers $n$.
The values form an arithmetic sequence because $f(n+1)-f(n)=m$ is constant for all integers $n$.
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 $n$.
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!
Let $f(x)=3x+2$. Over the equal interval $h=1$, which table of values and differences correctly demonstrates constant differences?
$f(0)=2,\ f(1)=5,\ f(2)=9,\ f(3)=14$ and differences: $3,4,5$ (constant).
$f(0)=2,\ f(1)=5,\ f(2)=8,\ f(3)=11$ and differences: $5-2=3,\ 8-5=3,\ 11-8=3$ (constant).
$f(0)=2,\ f(1)=5,\ f(2)=8,\ f(3)=11$ and ratios: $\frac{5}{2},\frac{8}{5},\frac{11}{8}$ (constant).
$f(0)=2,\ f(1)=6,\ f(2)=10,\ f(3)=14$ and differences: $4,4,4$ (constant).
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 f(x)=3x+2, the values are f(0)=2, f(1)=5, f(2)=8, f(3)=11, with differences 5-2=3, 8-5=3, 11-8=3, all constant at 3 (matching m=3 for h=1). Ratios like 5/2=2.5, 8/5=1.6 are not constant, as expected for linear. Choice A correctly demonstrates constant differences with the accurate table and calculations showing 3,3,3. Choice D fails by using the same table but claiming constant ratios, when in fact the ratios vary as 2.5, 1.6, 1.375. 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 an exponential function $g(x)=a\cdot b^x$ (with $a\neq 0$ and $b>0, b\neq 1$) when using any fixed equal interval $h$?
$\dfrac{g(x+h)}{g(x)}=a\cdot h$ for all $x$.
$g(x+h)-g(x)=b^h$ for all $x$.
$\dfrac{g(x+h)}{g(x)}=b^h$ for all $x$.
$g(x+h)-g(x)=a\cdot h$ 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 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 example, if $g(x)=5·2^x$ and h=1, then $g(x+1)/g(x)=[5·2^{x+1}$$]/(5·2^x$)=2, constant regardless of x. Differences like $g(x+1)-g(x)=5·2^{x+1}$ - $5·2^x$ = $5·2^x$ $(2-1)=5·2^x$ vary with x. Choice C correctly identifies that $g(x+h)/g(x)=b^h$ for all x, which is the defining property of constant ratios in exponentials. Choice A fails because it claims a constant difference of a·h, but as shown, the difference depends on x and is not constant 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!
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 geometric and comes from an exponential function.
The ratios 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 differences are constant, so it is arithmetic and comes from a linear 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!