Compare Linear and Exponential Growth - Algebra 2

A function is sampled at integer inputs, producing the sequence $4,\ 10,\ 25,\ 62.5$ . Over equal input intervals of $1$ , which statement correctly identifies the growth pattern and the most likely function type?
Answer: The ratios are constant: $\tfrac{10}{4}=\tfrac{25}{10}=\tfrac{62.5}{25}=2.5$ , so it matches exponential growth and corresponds to a geometric sequence. 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 4, 10, 25, 62.5, let's check both patterns. Differences: 10 - 4 = 6, 25 - 10 = 15, 62.5 - 25 = 37.5. The differences (6, 15, 37.5) are increasing - not constant! Ratios: 10/4 = 2.5, 25/10 = 2.5, 62.5/25 = 2.5. All ratios equal 2.5 - constant! Choice B correctly identifies constant ratios of 2.5, indicating exponential growth and a geometric sequence. This suggests the function is g(x) = 1.6· $(2.5)^x$ since g(0) = 1.6·1 = 1.6·2.5 = 4. Choice A correctly notes non-constant differences but wrongly concludes it must be linear anyway. Choice C incorrectly claims constant differences of +2.5 (the actual differences are 6, 15, 37.5). Choice D correctly identifies the ratios but wrongly associates constant ratios with linear growth instead of exponential growth. To determine if a function is linear or exponential from a table: calculate both differences and ratios - constant differences mean linear, constant ratios mean exponential. Why this matters: recognizing constant ratios tells you to multiply by 2.5 to get the next value: 62.5 × 2.5 = 156.25!
Which statement correctly connects function type to sequence type when sampling at integer inputs (equal intervals of $\Delta x=1$ )?
Answer: Sampling a linear function at integers produces an arithmetic sequence (constant differences), and sampling an exponential function produces a geometric sequence (constant ratios). 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! When you sample a linear function at integers (x = 0, 1, 2, 3, ...), the outputs have constant differences, forming an arithmetic sequence. For example, if f(x) = 3x + 2, then f(0) = 2, f(1) = 5, f(2) = 8, f(3) = 11, with constant differences of 3. When you sample an exponential function at integers, the outputs have constant ratios, forming a geometric sequence. For example, if g(x) = 2· $3^x$ , then g(0) = 2, g(1) = 6, g(2) = 18, g(3) = 54, with constant ratios of 3. Choice C correctly connects function type to sequence type: linear functions produce arithmetic sequences (constant differences) and exponential functions produce geometric sequences (constant ratios) when sampled at integer inputs. Choice A incorrectly claims linear functions produce geometric sequences, Choice B incorrectly claims exponential functions produce arithmetic sequences, and Choice D incorrectly states both produce arithmetic sequences. 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: $H(0)=5$ , $H(1)=10$ , $H(2)=20$ , $H(3)=40$ . Based on growth over equal intervals ( $\Delta x=1$ ), does this represent linear or exponential growth?
Answer: Exponential growth, because the ratios $\dfrac{H(1)}{H(0)}$ , $\dfrac{H(2)}{H(1)}$ , $\dfrac{H(3)}{H(2)}$ are all equal (geometric sequence). 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 analyze the given data: H(0) = 5, H(1) = 10, H(2) = 20, H(3) = 40. Calculate differences: H(1) - H(0) = 10 - 5 = 5, H(2) - H(1) = 20 - 10 = 10, H(3) - H(2) = 40 - 20 = 20. Differences are NOT constant (they double each time). Calculate ratios: H(1)/H(0) = 10/5 = 2, H(2)/H(1) = 20/10 = 2, H(3)/H(2) = 40/20 = 2. All ratios equal 2! Choice B correctly identifies this as exponential growth because the ratios are all equal (forming a geometric sequence), which is the defining characteristic of exponential functions. Choice A incorrectly claims the differences increase by a constant factor (that's not linear), Choice C incorrectly associates constant ratios with linear growth, and Choice D reverses the requirement (exponential needs constant ratios, not 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!
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?
Answer: City A will have approximately 65,000 people and City B will have approximately 41,472 people, for a difference of 23,528 people. 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.
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 $p(0),p(2),p(4),\dots$ ?
Answer: It shows constant differences over equal intervals, so $p(0),p(2),p(4),\dots$ is arithmetic and $p$ is consistent with linear growth. 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!
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: prove that for any real number $h$ , the ratio $\dfrac{g(x+h)}{g(x)}$ is constant (independent of $x$ ). Which expression correctly simplifies $\dfrac{g(x+h)}{g(x)}$ ?
Answer: $\dfrac{g(x+h)}{g(x)}=b^h$ (constant in $x$ when $h$ is fixed) 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· $b^x$ ] = a· $b^x$ · $b^h$ /( $a·b^x$ ) = $b^h$ , 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 this algebraically: g(x + h)/g(x) = (a· $b^{x+h}$ )/( $a·b^x$ ) = ( $a·b^x·b^h$ )/( $a·b^x$ ) = $b^h$ . The a and $b^x$ terms cancel completely, leaving only $b^h$ , which is constant with respect to x. Choice A correctly identifies that exponential functions have constant ratios of $b^h$ through proper algebra. Choice B incorrectly keeps the coefficient a in the ratio (it should cancel), Choice C incorrectly suggests the ratio is $b^{x+h}$ (missing the division), and Choice D confuses the base and exponent ( $h^b$ instead of $b^h$ ). 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 over equal unit intervals ( $h=1$ ) that the ratios are constant by computing $g(0), g(1), g(2), g(3)$ and the consecutive ratios. Which option correctly demonstrates the constant-ratio property and identifies the associated sequence type at integer inputs?
Answer: $g(0)=2, g(1)=6, g(2)=18, g(3)=54$ ; ratios: $3,3,3$ (constant). So $\{g(n)\}$ is geometric. 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 compute g(x) = 2· $3^x$ at x = 0, 1, 2, 3: 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 ratios are: 6/2 = 3, 18/6 = 3, 54/18 = 3. All ratios equal 3, which is the base b! Choice A correctly computes these values and ratios, showing they're constant at 3, and correctly identifies that {g(n)} forms a geometric sequence. Choice B shows values for a linear function (2, 5, 8, 11) with constant differences of 3, not the exponential function g(x) = 2· $3^x$ . Choice C starts correctly with g(0) = 2 and g(1) = 6, but then incorrectly computes g(2) = 12 instead of 18, leading to non-constant ratios. Choice D correctly computes the values but then incorrectly calculates differences instead of ratios - for an exponential function, differences are not constant (they grow: 4, 12, 36). 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!
Demonstrate that the linear function $f(x)=3x+2$ has constant differences over unit intervals. Which statement correctly shows the constant difference for $\Delta x=1$ ?
Answer: $f(x+1)-f(x)=3$ for all $x$ , so the outputs form an arithmetic sequence when sampled at integers. 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 f(x) = 3x + 2 with h = 1, let's calculate: f(x + 1) - f(x) = [3(x + 1) + 2] - [3x + 2] = 3x + 3 + 2 - 3x - 2 = 3. This shows that no matter what x value you start at, moving right by 1 unit always adds 3 to the output. We can verify: f(0) = 2, f(1) = 5, f(2) = 8, f(3) = 11, and the differences are 5-2=3, 8-5=3, 11-8=3. Choice A correctly proves that f(x+1) - f(x) = 3 for all x, demonstrating constant differences and showing the outputs form an arithmetic sequence. Choice B incorrectly suggests constant ratios (that would be exponential), Choice C incorrectly claims f(x+1) - f(x) = x + 3 (the x should cancel out), and Choice D incorrectly states the difference is 5 (that's f(1), not the difference). 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 $h$ ?
Answer: $f(x+h)-f(x)=m\cdot h$ for all $x$ . 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!
For the linear function $f(x)= -4x+9$ , analyze growth over equal intervals of length $h=2$ . What is $f(x+2)-f(x)$ , and what does it show?
Answer: $f(x+2)-f(x)=-8$ , so the function has constant differences over intervals of length 2 (arithmetic pattern). 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 $f(x) = -4x + 9$ with $h = 2$ , let's calculate: $f(x + 2) - f(x) = [-4(x + 2) + 9] - [-4x + 9] = -4x - 8 + 9 - (-4x) - 9 = -4x - 8 + 9 + 4x - 9 = -8$ . The $x$ terms cancel and constants simplify to $-8$ . We can verify: $f(0) = 9$ , $f(2) = 1$ , $f(4) = -7$ , and differences are $1-9=-8$ , $(-7)-1=-8$ . Choice B correctly demonstrates that $f(x+2) - f(x) = -8$ , showing constant differences over intervals of length 2 and confirming the arithmetic pattern. Choice A incorrectly calculates the difference as $-4$ (that would be for $h=1$ ), Choice C incorrectly keeps $x$ in the expression (it should cancel), and Choice D incorrectly suggests this is a ratio (ratios are for exponential functions). 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: $y_2 - y_1$ , $y_3 - y_2$ , $y_4 - y_3$ , (3) calculate ratios: $y_2/y_1$ , $y_3/y_2$ , $y_4/y_3$ , (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?
Answer: The values form an arithmetic sequence because $f(n+1)-f(n)=m$ is constant for all integers $n$ . 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!
For the linear function $f(x)= -2x+7$ , show that over equal intervals of length $h=3$ , the change in function value is constant by simplifying $f(x+3)-f(x)$ . Which expression is correct?
Answer: $f(x+3)-f(x)=[-2(x+3)+7]-[-2x+7]=-6$ , constant in $x$ . 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! Let's verify algebraically for f(x) = -2x + 7 with h = 3: f(x+3) - f(x) = [-2(x+3) + 7] - [-2x + 7] = [-2x - 6 + 7] - [-2x + 7] = -2x + 1 - (-2x + 7) = -2x + 1 + 2x - 7 = -6. The x terms cancel, leaving -6, which is constant with respect to x. Note that -6 = (-2)(3) = mh, as expected. Choice A correctly shows this calculation, demonstrating that the difference equals -6 and is constant in x. Choice B makes an error by not distributing the negative sign properly when subtracting [-2x + 7], leading to an expression -2x - 6 that incorrectly depends on x. Choice C incorrectly attempts to find a ratio (using division) instead of a difference (using subtraction) for a linear function. Choice D has the wrong sign - it gets +6 instead of -6, likely by making a sign error when distributing or subtracting. 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!
Examine the coordinate plane shown. Function $f$ passes through points $(1,5)$ , $(3,11)$ , and $(5,17)$ . Function $g$ passes through points $(0,4)$ , $(2,16)$ , and $(4,64)$ . Which analysis of their growth patterns over equal intervals is most accurate?
Answer: Function $f$ has constant first differences of 3 indicating linear growth, while function $g$ has constant ratios of 4 indicating exponential growth. For function $f$ : The differences between consecutive y-values are $11-5=6$ (from x=1 to x=3) and $17-11=6$ (from x=3 to x=5). Since these are over intervals of length 2, the rate of change is $\frac{6}{2}=3$ per unit, indicating linear growth with slope 3. For function $g$ : Over intervals of length 2, ratios are $\frac{16}{4}=4$ and $\frac{64}{16}=4$ , indicating exponential growth with factor 4 every 2 units. Choice A incorrectly identifies $f$ as quadratic and $g$ as having base 2. Choice C states the difference as 6 per unit instead of 3 per unit. Choice D incorrectly identifies $f$ as exponential.
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?
Answer: The bacteria demonstrates exponential growth reaching 2700 organisms, while the plant shows linear growth reaching 15.5 cm in height. 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.
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.
Answer: $g(0)=2,g(1)=6,g(2)=18,g(3)=54$ ; ratios are $3,3,3$ , so the growth is exponential and the values form a geometric sequence. 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!
Prove that any linear function $f(x)=mx+b$ grows by equal differences over equal intervals. Specifically, show that for any real number $h$ , the difference $f(x+h)-f(x)$ is constant with respect to $x$ , and explain how this implies that the values of $f(n)$ at integer inputs form an arithmetic sequence.
Answer: $f(x+h)-f(x)=[m(x+h)+b]-[mx+b]=mh$ , which depends only on $h$ (constant in $x$ ). Therefore $f(n+1)-f(n)=m$ is constant, so $\{f(n)\}$ is arithmetic. 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! Let's verify: f(x+h) - f(x) = m(x+h) + b - (mx + b) = mx + mh + b - mx - b = mh. The x terms cancel, leaving only mh, which is constant with respect to x. Choice A correctly proves that f(x+h) - f(x) = mh is constant in x and recognizes that when we sample at integers (h=1), we get f(n+1) - f(n) = m, making {f(n)} an arithmetic sequence with common difference m. Choice B makes an algebra error by writing -mx + b instead of -(mx + b), leading to an incorrect expression mx + mh + 2b. Choice C incorrectly attempts to find a ratio instead of a difference. Choice D correctly starts the calculation but then claims the difference depends on x (it doesn't - the mx terms cancel) and incorrectly states linear 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!
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?
Answer: This shows exponential growth because consecutive ratios are constant; the values form a geometric sequence. 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!
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.
Answer: 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 $x$ . 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!
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?
Answer: 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. 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.
Prove that any linear function $f(x)=mx+b$ grows by equal differences over equal intervals. Specifically, show that for any real number $h$ , the difference $f(x+h)-f(x)$ is constant (does not depend on $x$ ), and state what constant it equals. Also connect this to why the values $f(0),f(1),f(2),\dots$ form an arithmetic sequence.
Answer: $f(x+h)-f(x)=[m(x+h)+b]-[mx+b]=mh$ , which is constant for fixed $h$ ; therefore $f(0),f(1),f(2),\dots$ has constant difference $m$ and is arithmetic. 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] = mx + mh + 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! Choice A correctly proves that f(x+h) - f(x) = mh through proper algebraic simplification, showing the difference is constant and equals mh. Choice B incorrectly claims the difference equals mx + mh, failing to cancel the mx terms. Choice C incorrectly attempts to find a ratio instead of a difference. Choice D incorrectly claims the difference equals b, missing the mh term entirely. 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), (2) calculate differences between consecutive y-values, (3) if differences are constant → linear (slope = that constant difference per unit interval). Since f(x+1) - f(x) = m for a linear function, the values f(0), f(1), f(2),... form an arithmetic sequence with common difference m, perfectly connecting the algebraic property to the sequence type!

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