Algebra Quiz: Recognize Constant Rate Changes

What this quiz covers

This quiz focuses on Recognize Constant Rate Changes, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

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

Question 1

1. Yes; constant rate $\Delta y/\Delta x = 3$ (the $y$-values increase by 3 each time $x$ increases by 1). (correct answer)
2. Yes; constant rate $\Delta y/\Delta x = 12$ (because $17-5=12$).
3. No; the rate is not constant because $y$ is increasing.
4. Yes; constant rate $\Delta y/\Delta x = 2$ (the $y$-values increase by 2 each time).

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. To check if a rate is constant from a table: calculate Δy/Δx (change in y over change in x) for each consecutive pair of points. If you get the same number every time, the rate is constant. If the values differ, the rate is non-constant. Example: if differences are 3, 3, 3, 3—constant! If differences are 2, 4, 6, 8—non-constant (actually quadratic pattern). Let's check if the rate is constant by calculating Δy/Δx for each interval in the table: From x = 0 to x = 1: Δy/Δx = (8 - 5)/(1 - 0) = 3/1 = 3. From x = 1 to x = 2: Δy/Δx = (11 - 8)/(2 - 1) = 3/1 = 3. From x = 2 to x = 3: Δy/Δx = (14 - 11)/(3 - 2) = 3/1 = 3. From x = 3 to x = 4: Δy/Δx = (17 - 14)/(4 - 3) = 3/1 = 3. All rates equal 3, so yes, constant rate of 3! Choice A correctly identifies the rate as constant with Δy/Δx = 3 because the y-values increase by 3 for each 1-unit increase in x, showing equal differences. Choice D says constant rate of 12, but that's just the total change from first to last without dividing by the intervals—it's easy to forget to calculate per interval, but always divide Δy by Δx for each pair! The foolproof test for constant rate from a table: (1) Make sure your x-values increase by the same amount each time (like going 1, 2, 3, 4 or 0, 5, 10, 15), (2) Calculate the differences in y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, etc., (3) If all differences are equal, rate is constant! If they differ, rate is not constant. This works every time with equally-spaced x-values. Common confusion: constant ratio ≠ constant rate! Geometric sequences have constant ratio (multiply by same factor), but their rate of change is not constant—it's increasing. Example: 2, 6, 18, 54... has constant ratio (×3) but rate goes 4, 12, 36 (not constant). Constant rate means linear, constant ratio means exponential!

Question 2

1. No; the flat fee means the rate of change cannot be constant.
2. Yes; constant rate $=4$ because the cost starts at \4$when miles are$0$.
3. Yes; constant rate $=2$ because the cost increases by \2$ for each additional mile (linear). (correct answer)
4. Yes; constant rate $=6$ because $4+2=6$.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. In real-world contexts, constant rate sounds like: 'travels at steady 60 mph,' 'costs 5 per item,' 'fills at 10 gallons per minute'—the 'per' language and steady/constant/fixed words signal constant rate. Non-constant rate sounds like: 'accelerating,' 'slowing down,' 'doubling each hour,' 'speed increasing'—these signal that the rate itself is changing! In this context, 'a taxi charges a flat fee of 4 plus $2 for each mile traveled,' we analyze: the language 'plus$2 for each mile' indicates constant rate. Each mile produces the same change in cost, specifically $2 per mile. Choice A correctly identifies the rate as constant at 2 because the cost increases by$2 for each additional mile, showing the linear nature of the relationship. Choice B says the flat fee means the rate cannot be constant, but the flat fee is just the y-intercept in a linear function, which doesn't affect the constancy of the rate—the rate is still constant as long as the per-mile charge is fixed. It's easy to think intercepts disrupt constancy, but they don't; only changes in the per-unit amount would make it non-constant! Context language decoder: words like 'constant speed,' 'steady rate,' '$X per unit,' 'every hour the same amount' → constant rate (linear). Words like 'accelerating,' 'percent per year,' 'doubling,' 'slowing down,' 'squared' → non-constant rate (nonlinear). The language almost always reveals which type!

Question 3

1. No; because $w$ increases by 1 each time, the rate must be zero.
2. Yes; the differences in $h$ are $2,3,4,5$, which shows a constant rate.
3. No; the differences in $h$ for each 1-week increase in $w$ are not all the same, so the rate is non-constant. (correct answer)
4. Yes; the ratio $h/w$ is constant, so the rate of change is constant.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. To check if a rate is constant from a table: calculate Δy/Δx (change in y over change in x) for each consecutive pair of points. If you get the same number every time, the rate is constant. If the values differ, the rate is non-constant. Example: if differences are 3, 3, 3, 3—constant! If differences are 2, 4, 6, 8—non-constant (actually quadratic pattern). Let's check if the rate is constant by calculating Δh/Δw for each interval in the table: From w = 0 to w = 1: Δh/Δw = (2 - 0)/(1 - 0) = 2/1 = 2. From w = 1 to w = 2: Δh/Δw = (5 - 2)/(2 - 1) = 3/1 = 3. From w = 2 to w = 3: Δh/Δw = (9 - 5)/(3 - 2) = 4/1 = 4. From w = 3 to w = 4: Δh/Δw = (14 - 9)/(4 - 3) = 5/1 = 5. The rates are different (2, 3, 4, 5), so no, the rate is not constant—it's changing. Choice B correctly identifies the rate as non-constant because the differences in h for each 1-week increase are not all the same (they're 2, 3, 4, 5—an increasing pattern). Choice C sees the pattern 2, 3, 4, 5 and thinks this shows constant rate, but a constant rate means the SAME number repeated, not a pattern of different numbers. To confirm constant rate, all differences must be identical! The foolproof test for constant rate from a table: (1) Make sure your x-values increase by the same amount each time (like going 1, 2, 3, 4 or 0, 5, 10, 15), (2) Calculate the differences in y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, etc., (3) If all differences are equal, rate is constant! If they differ, rate is not constant. This works every time with equally-spaced x-values.

Question 4

1. Yes; the rate is constant because the balance is always increasing.
2. No; the rate is non-constant because $B/w$ is not constant.
3. Yes; the rate is constant because $w$ increases by 1 each time.
4. No; the rate is non-constant because the weekly changes in $B$ are not all equal. (correct answer)

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. To check if a rate is constant from a table: calculate Δy/Δx (change in y over change in x) for each consecutive pair of points. If you get the same number every time, the rate is constant. If the values differ, the rate is non-constant. Example: if differences are 3, 3, 3, 3—constant! If differences are 2, 4, 6, 8—non-constant (actually quadratic pattern). Let's check if the rate is constant by calculating ΔB/Δw for each interval in the table: From w = 0 to w = 1: ΔB/Δw = (110 - 100)/(1 - 0) = 10/1 = 10. From w = 1 to w = 2: ΔB/Δw = (125 - 110)/(2 - 1) = 15/1 = 15. From w = 2 to w = 3: ΔB/Δw = (135 - 125)/(3 - 2) = 10/1 = 10. From w = 3 to w = 4: ΔB/Δw = (150 - 135)/(4 - 3) = 15/1 = 15. The rates are different (10, 15, 10, 15), so no, the rate is not constant—it's changing. Choice B correctly identifies the rate as non-constant because the weekly changes in B are not all equal (they alternate between 10 and 15). Choice A says the rate is constant because the balance is always increasing, but constant rate requires the AMOUNT of increase to be the same each time, not just that it increases. To confirm constant rate, you need to check multiple intervals—if even one differs, it's non-constant! Always verify across at least 3-4 intervals before concluding constancy. The foolproof test for constant rate from a table: (1) Make sure your x-values increase by the same amount each time (like going 1, 2, 3, 4 or 0, 5, 10, 15), (2) Calculate the differences in y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, etc., (3) If all differences are equal, rate is constant! If they differ, rate is not constant. This works every time with equally-spaced x-values.

Question 5

1. Yes; constant rate $\Delta d/\Delta t = 360$ miles per hour.
2. Yes; constant rate $\Delta d/\Delta t = 60$ miles per hour. (correct answer)
3. No; the rate is not constant because the distances are different each time.
4. Yes; constant rate $\Delta d/\Delta t = 120$ miles per hour.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. To check if a rate is constant from a table: calculate $\Delta y / \Delta x$ (change in y over change in x) for each consecutive pair of points. If you get the same number every time, the rate is constant. If the values differ, the rate is non-constant. Example: if differences are 3, 3, 3, 3—constant! If differences are 2, 4, 6, 8—non-constant (actually quadratic pattern). Let's check if the rate is constant by calculating $\Delta y / \Delta x$ for each interval in the table: From $t = 0$ to $t = 2$: $\Delta d / \Delta t = (120 - 0)/(2 - 0) = 120/2 = 60$. From $t = 2$ to $t = 4$: $\Delta d / \Delta t = (240 - 120)/(4 - 2) = 120/2 = 60$. From $t = 4$ to $t = 6$: $\Delta d / \Delta t = (360 - 240)/(6 - 4) = 120/2 = 60$. All rates equal 60, so yes, constant rate of 60! Choice C correctly identifies the rate as constant with $\Delta d / \Delta t = 60$ because after dividing the equal $\Delta d$ (120) by $\Delta t$ (2), we get consistent 60 mph across intervals. Choice B says yes with 120, but that's forgetting to divide by $\Delta t=2$—it's easy to just look at $\Delta d$ without the 'per hour' part; always compute the full ratio $\Delta y / \Delta x$! The foolproof test for constant rate from a table: (1) Make sure your x-values increase by the same amount each time (like going 1, 2, 3, 4 or 0, 5, 10, 15), (2) Calculate the differences in y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, etc., (3) If all differences are equal, rate is constant! If they differ, rate is not constant. This works every time with equally-spaced x-values. If x-intervals aren't equal, you must divide each $\Delta y$ by its $\Delta x$ to check if the ratios are equal—that's key for non-uniform spacing!

Question 6

1. Yes; constant rate $\Delta C/\Delta v = 4$ dollars per visit. (correct answer)
2. No; it is not constant because there is a $25$ dollar membership fee.
3. Yes; constant rate $\Delta C/\Delta v = 25$ dollars per visit.
4. No; the rate changes as $v$ increases because the total cost increases.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. In real-world contexts, constant rate sounds like: 'travels at steady 60 mph,' 'costs 5 per item,' 'fills at 10 gallons per minute'—the 'per' language and steady/constant/fixed words signal constant rate. Non-constant rate sounds like: 'accelerating,' 'slowing down,' 'doubling each hour,' 'speed increasing'—these signal that the rate itself is changing! In this context, 'a gym charges a membership fee plus a fixed cost per visit with C = 25 + 4v,' we analyze: the language 'fixed cost per visit' indicates constant rate. Each unit of input (visit) produces the same change in output (cost), specifically 4 dollars per visit. Choice A correctly identifies the rate as constant with ΔC/Δv = 4 because the 'per visit' term is fixed, and the membership fee is just a starting point that doesn't affect the rate of change. Choice B says no because of the 25 membership fee, but that's a supportive reminder—the fee is like the y-intercept in y=mx+b, which doesn't change the constant slope m; it's easy to think constants make it nonlinear, but they don't! Context language decoder: words like 'constant speed,' 'steady rate,' '$X per unit,' 'every hour the same amount' → constant rate (linear). Words like 'accelerating,' 'percent per year,' 'doubling,' 'slowing down,' 'squared' → non-constant rate (nonlinear). The language almost always reveals which type! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 7

1. No; the rate is not constant because $g(0)=0$.
2. Yes; constant rate of change $= 2$ because the coefficient of $x$ is 2.
3. Yes; constant rate of change $= 1$ because the leading coefficient is 1.
4. No; it is nonlinear (quadratic), so $\Delta y/\Delta x$ is not constant. (correct answer)

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. A constant rate of change means that for every unit increase in x, y changes by the same amount every time—this constant rate is exactly what makes a function linear! Looking at the function g(x) = x² + 2x: This is a quadratic function (contains x²), and quadratic functions have variable rates of change—the rate is different at different x-values, so it's not constant. The presence of x² is the key indicator that this is nonlinear. Choice C correctly identifies the rate as non-constant because it recognizes this is a nonlinear (quadratic) function where Δy/Δx varies. Choice A incorrectly focuses on the coefficient 2 of the linear term 2x, but the presence of x² makes the entire function quadratic with non-constant rate—you can't ignore the x² term! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 8

1. Yes; the constant rate is $50$ dollars per month because $1.5-1=0.5$.
2. Yes; it increases by a constant amount each month because the base is $1.5$.
3. No; it is exponential, so it has a constant ratio but not a constant difference (nonlinear). (correct answer)
4. No; but it is linear since it starts at $100$.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. A constant rate of change means that for every unit increase in x, y changes by the same amount every time: if going from x = 1 to x = 2 increases y by 5, and going from x = 2 to x = 3 also increases y by 5, and this pattern continues, then the rate is constant at 5 units per unit. Looking at the function B(t) = 100·(1.5)^t: This is an exponential function, and exponential functions have variable rates of change—the rate is different at different t-values, so it's not constant. Choice B correctly identifies that it is exponential, so it has a constant ratio but not a constant difference, making it nonlinear. Choice A confuses constant ratio with constant rate: in exponential contexts, consecutive terms have a constant ratio (multiply by 1.5 each time), but the rate of change (ΔB/Δt) is increasing. Common confusion: constant ratio ≠ constant rate! Geometric sequences have constant ratio (multiply by same factor), but their rate of change is not constant—it's increasing. For B(t) = 100·(1.5)^t: B(0)=100, B(1)=150, B(2)=225, B(3)=337.5... The differences are 50, 75, 112.5 (not constant). Constant ratio means exponential, constant rate means linear—don't mix these up!

Question 9

1. Yes; the constant rate is $-1$ because $-1$ is the y-intercept.
2. No; but it would be constant because the second differences are constant.
3. Yes; the constant rate is $3$ because the coefficient of $x^2$ is $3$.
4. No; it is nonlinear (quadratic), so $\Delta y/\Delta x$ is not constant. (correct answer)

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. Linear functions are the ONLY functions with constant rates of change: if a graph is a straight line, the rate is constant. If the graph curves (like a parabola or exponential curve), the rate is changing. Looking at the function f(x) = 3x² - 1: This is a quadratic function, and quadratic functions have variable rates of change—the rate is different at different x-values, so it's not constant. Choice B correctly identifies that it is nonlinear (quadratic), so Δy/Δx is not constant because quadratic functions create parabolas, which curve, meaning the steepness changes at every point. Choice A says the constant rate is 3 because that's the coefficient of x², but this confuses the coefficient with the rate of change—in quadratics, the rate of change varies and is not equal to any single coefficient. Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 10

1. Yes; constant rate of change $=2$ because the coefficient of $x$ is 2.
2. No; it is nonlinear (quadratic), so $\Delta g/\Delta x$ is not constant. (correct answer)
3. Yes; constant rate of change $=1$ because the function increases by 1 when $x$ increases by 1.
4. No; it has a constant rate of change only when $x=0$.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. Linear functions are the ONLY functions with constant rates of change: if a function has x² or any power other than 1, it's not linear and doesn't have a constant rate. Looking at the function g(x) = 2x² + 1: This is a quadratic function (has x²), and quadratic functions have variable rates of change—the rate is different at different x-values, so it's not constant. The parabola gets steeper as you move away from the vertex, meaning the rate of change increases. Choice B correctly identifies this as nonlinear (quadratic) with non-constant Δg/Δx, recognizing that the x² term makes it impossible to have constant rate. Choice A sees the coefficient 2 but misses that it's attached to x², not just x—the form matters more than the number! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 = constant rate, any other power = non-constant rate.

Question 11

1. The rate of change is not constant because the function has a negative coefficient.
2. The rate of change is constant and equals $-4$ because $g(x)$ is linear of the form $mx+b$. (correct answer)
3. The rate of change is constant and equals $7$ because $g(0)=7$.
4. The rate of change is constant and equals $4$ because the function decreases by 4 when $x$ decreases by 1.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. Linear functions are the ONLY functions with constant rates of change: if a graph is a straight line, the rate is constant. If the graph curves (like a parabola or exponential curve), the rate is changing. You can visually spot constant rate—it's straightness! A steeper line has larger constant rate, flatter line has smaller constant rate, but both are constant as long as the line is straight. Looking at the function g(x) = 7 - 4x: This is a linear function in the form y = mx + b with slope m = -4, which means the rate of change is constant at -4 everywhere. We can verify: for any 1-unit increase in x, g(x) decreases by 4. For example, g(0) = 7, g(1) = 3, g(2) = -1, showing consistent decreases of 4. Choice B correctly states the rate of change is constant and equals -4 because g(x) is linear of the form mx + b, where the coefficient of x (which is -4) is the constant rate of change. Choice C confuses the y-intercept g(0) = 7 with the rate of change—the rate is the slope (-4), not where the line crosses the y-axis. Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 12

1. No; the rate is not constant because $x$ changes.
2. Yes; constant rate of change $= 7$ because the function is linear of the form $y = mx + b$. (correct answer)
3. Yes; constant rate of change $= -4$ because $-4$ is the constant term.
4. No; the rate is not constant because the function includes a $-4$.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. A constant rate of change means that for every unit increase in x, y changes by the same amount every time: if going from x = 1 to x = 2 increases y by 5, and going from x = 2 to x = 3 also increases y by 5, and this pattern continues, then the rate is constant at 5 units per unit. This constant rate is exactly what makes a function linear—it's the slope! Looking at the function f(x) = 7x - 4: This is a linear function in the form y = mx + b with slope m = 7, which means the rate of change is constant at 7 everywhere. The -4 is just the y-intercept and doesn't affect the rate of change. Choice A correctly identifies the rate as constant at 7 because it recognizes this is a linear function where the coefficient of x (which is 7) represents the constant rate of change. Choice C confuses the constant term -4 with the rate of change, but the rate comes from the coefficient of x, not the constant term. Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 13

1. No; it is nonlinear because it has a constant term (+9).
2. No; the rate changes because $x$ changes.
3. Yes; constant rate $\Delta y/\Delta x = 9$.
4. Yes; constant rate $\Delta y/\Delta x = -4$ (it is linear of the form $y=mx+b$). (correct answer)

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. A constant rate of change means that for every unit increase in x, y changes by the same amount every time: if going from x = 1 to x = 2 increases y by 5, and going from x = 2 to x = 3 also increases y by 5, and this pattern continues, then the rate is constant at 5 units per unit. This constant rate is exactly what makes a function linear—it's the slope! Looking at the function f(x) = -4x + 9: This is a linear function in the form y = mx + b with slope m = -4, which means the rate of change is constant at -4 everywhere. Choice C correctly identifies the rate as constant with Δy/Δx = -4 because it is linear of the form y=mx+b, where m is the unchanging slope. Choice A says no because it is nonlinear due to the constant term +9, but that's a common mix-up—the +b term is the y-intercept and doesn't affect the rate; only nonlinear terms like x² would make the rate vary! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate. Linear functions are the ONLY functions with constant rates of change: if a graph is a straight line, the rate is constant. If the graph curves (like a parabola or exponential curve), the rate is changing. You can visually spot constant rate—it's straightness! A steeper line has larger constant rate, flatter line has smaller constant rate, but both are constant as long as the line is straight.

Question 14

1. No; it is nonlinear because it has a subtraction.
2. Yes; constant rate of change $=4$ because it is in the form $y=mx+b$. (correct answer)
3. Yes; constant rate of change $=-7$ because $b=-7$.
4. No; the rate changes as $x$ increases because $x$ is multiplied.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. Linear functions are the ONLY functions with constant rates of change: if a graph is a straight line, the rate is constant. A linear function in the form y = mx + b has a constant rate of change equal to m, the coefficient of x. Looking at the function f(x) = 4x - 7: This is a linear function in the form y = mx + b with slope m = 4, which means the rate of change is constant at 4 everywhere. For every 1-unit increase in x, f(x) increases by 4 units, regardless of where you are on the line. Choice B correctly identifies that this is a linear function with constant rate of change = 4 because it recognizes the y = mx + b form where m = 4. Choice C confuses the y-intercept (b = -7) with the rate of change—the constant term tells you where the line crosses the y-axis, not how steep it is! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 15

1. The water volume changes at a constant rate of 11.5 gallons per minute throughout the entire process
2. The water volume changes at a constant rate during each interval, but the overall process does not have a constant rate (correct answer)
3. The water volume changes at a constant rate of 23 gallons per minute when considering the combined flow rates
4. The water volume does not change at a constant rate during any part of the filling process due to varying conditions

Explanation: The rate of change is constant within each time interval (15 gal/min for 0-20 min, 8 gal/min for 20-50 min), but the overall process has two different constant rates, making the entire process non-linear. A is incorrect because it averages the rates incorrectly. C is wrong because you don't add the rates. D is incorrect because each individual interval does have a constant rate.

Question 16

1. Colony A maintains a constant growth rate, while Colony B's growth rate increases exponentially over the time period (correct answer)
2. Both colonies exhibit constant rates of change, but Colony A grows faster than Colony B throughout the observation
3. Colony A has a variable growth rate that averages 50 per hour, while Colony B has a constant doubling rate
4. Neither colony demonstrates a constant rate of change due to the biological nature of population growth patterns

Explanation: Colony A increases by exactly 50 bacteria each hour (linear: 200, 250, 300, 350, 400, 450, 500), showing constant rate of change. Colony B doubles every 2 hours (exponential: 150, 150, 300, 300, 600, 600, 1200), which means its rate of change increases over time. B incorrectly claims both are constant. C incorrectly describes A as variable. D incorrectly rejects A's constant rate.

Question 17

1. The water level decreases at a constant rate of 4 inches per hour when considering the average across both phases
2. The water level decreases at variable rates throughout the draining process due to changing pipe pressure conditions
3. The water level decreases at constant rates during each phase: 6 inches/hour initially, then 2 inches/hour subsequently (correct answer)
4. The water level decreases at an increasing rate as the pool empties and gravitational effects become stronger

Explanation: Both functions are linear within their respective time periods. h(t) = 48 - 6t has a constant rate of -6 inches/hour for the first phase, and h(t) = 24 - 2t has a constant rate of -2 inches/hour for the second phase. A incorrectly averages the rates. B incorrectly claims variable rates within phases. D incorrectly describes an accelerating rate.

Question 18

1. Both the radius and volume of the balloon increase at constant rates throughout the inflation process
2. Both the radius and volume increase at variable rates that depend on the initial balloon size
3. The radius increases at a variable rate, while the volume increases at a constant rate of $\frac{4}{3}\pi$ per second
4. The radius increases at a constant rate of 2 cm/sec, while the volume increases at a variable rate (correct answer)

Explanation: When analyzing rates of change in function problems, you need to distinguish between constant and variable rates by examining how quickly each quantity changes over time. Let's examine each function separately. The radius function is $r(t) = 2t + 3$. Since this is a linear function, the rate of change is the coefficient of $t$, which is 2. This means the radius increases at a constant rate of 2 cm per second throughout the entire inflation process. For volume, we have $V = \frac{4}{3}\pi r^3$. Substituting the radius function: $V = \frac{4}{3}\pi (2t + 3)^3$. Since volume depends on the cube of the radius, and the radius is changing, the volume's rate of change varies over time. As the balloon gets larger, the same increase in radius produces a much larger increase in volume. Looking at the answer choices: Choice A is incorrect because while radius increases at a constant rate, volume does not. Choice B is wrong because the radius rate doesn't depend on initial size—it's always 2 cm/sec regardless of when you measure it. Choice C reverses the situation, incorrectly stating that radius has a variable rate and volume has a constant rate of $\frac{4}{3}\pi$ (which isn't even dimensionally correct for a rate). Choice D correctly identifies that radius increases at a constant 2 cm/sec while volume increases at a variable rate. Remember: linear functions always have constant rates of change, while functions involving powers (like cubes) typically have variable rates of change.

Question 19

1. Both machines produce widgets at constant rates, with the second machine having a higher constant rate of production
2. The first machine produces at a constant rate of 45 widgets per hour, while the second machine's rate varies with time (correct answer)
3. The first machine's rate decreases over time, while the second machine maintains a constant rate of 60 widgets per hour
4. Both machines have variable rates of production, but the first machine's rate changes more gradually than the second's

Explanation: The first function W(t) = 45t + 120 is linear with a constant rate of change of 45 widgets/hour. The second function W(t) = 60t² - 15t + 50 is quadratic, so its rate of change (derivative = 120t - 15) varies with time. A incorrectly claims both are constant. C incorrectly describes the rates. D incorrectly claims the first machine has a variable rate.

Question 20

1. Yes; constant rate of $25$ \/month because $\Delta C/\Delta m = 25$ for each 1-month interval. (correct answer)
2. Yes; constant rate of $5$ \/month because $\Delta C$ is $5$ from $m=0$ to $m=1$.
3. No; the change in $C$ is not the same for each 1-month increase in $m$, so the rate is non-constant.
4. Yes; constant rate of $20$ \/month because $C/m$ stays the same.

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. To check if a rate is constant from a table: calculate Δy/Δx (change in y over change in x) for each consecutive pair of points. If you get the same number every time, the rate is constant. If the values differ, the rate is non-constant. Example: if differences are 3, 3, 3, 3—constant! If differences are 2, 4, 6, 8—non-constant (actually quadratic pattern). Let's check if the rate is constant by calculating ΔC/Δm for each interval in the table: From m = 0 to m = 1: ΔC/Δm = (30 - 5)/(1 - 0) = 25/1 = 25. From m = 1 to m = 2: ΔC/Δm = (55 - 30)/(2 - 1) = 25/1 = 25. From m = 2 to m = 3: ΔC/Δm = (80 - 55)/(3 - 2) = 25/1 = 25. All rates equal 25, so yes, constant rate of 25 $/month! Choice B correctly identifies the rate as constant because showing equal differences of 25 for each 1-month interval. Choice C calculates C/m (total/months) instead of ΔC/Δm (change/change)—this gives average cost per month from start, not the rate of change. To confirm constant rate, you need to check multiple intervals—if even one differs, it's non-constant! Always verify across at least 3-4 intervals before concluding constancy. The foolproof test for constant rate from a table: (1) Make sure your x-values increase by the same amount each time (like going 1, 2, 3, 4 or 0, 5, 10, 15), (2) Calculate the differences in y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, etc., (3) If all differences are equal, rate is constant! If they differ, rate is not constant. This works every time with equally-spaced x-values.