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Recognize Constant Rate Changes - Algebra 1

  1. Three investment accounts are tracked over time. Account X gains $150 each month. Account Y gains $100 the first month, then the monthly gain increases by $25 each subsequent month. Account Z gains $200 every other month and $0 on alternate months. Which account(s) show a constant rate of change?
    Answer: Only Account X demonstrates a constant rate of change at $150 per month throughout the investment period Account X gains $150 each month, which is a constant rate. Account Y has increasing monthly gains ($100, $125, $150, ...), so its rate increases over time. Account Z alternates between $200 and $0, which creates a variable rate (not constant even when averaged). B incorrectly includes Z. C incorrectly describes Y as constant. D incorrectly claims all are constant.
  2. Over a 6-hour observation period, which statement best describes the rate of change in population for each colony?
    Answer: Colony A maintains a constant growth rate, while Colony B's growth rate increases exponentially over the time period 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.
  3. A tank is being filled at a steady rate. After 2 minutes it contains 9 liters, and after 6 minutes it contains 21 liters. What is the rate of change in liters per minute, and is it constant?
    Answer: Rate =3=3 L/min, and it is constant because ΔV/Δt=(21−9)/(6−2)=3\Delta V/\Delta t = (21-9)/(6-2)=3. 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. In this context, 'a tank is being filled at a steady rate,' we analyze: the language 'steady rate' indicates constant rate. Let's calculate: From 2 to 6 minutes (Δt = 4), the volume goes from 9 to 21 liters (ΔV = 12). Rate = ΔV/Δt = 12/4 = 3 liters per minute. Since it's filling at a 'steady rate,' this 3 L/min is constant throughout. Choice B correctly calculates the rate as 3 L/min using ΔV/Δt = (21-9)/(6-2) = 12/4 = 3, and recognizes that 'steady' means constant. Choice A makes a calculation error by not dividing the change in volume by the change in time—rate requires division to get the 'per minute' unit! Context language decoder: words like 'constant speed,' 'steady rate,' 'X per unit,' 'every hour the same amount' → constant rate (linear). The calculation ΔV/Δt gives you the exact constant rate value.
  4. A coordinate plane shows a line passing through the points (0,2)(0,2) and (4,10)(4,10). Does this graph represent a constant rate of change? If yes, what is the rate (slope) Δy/Δx\Delta y/\Delta x?
    Answer: Yes; constant rate 22 because 10−2=810-2=8 and 8/4=28/4=2. 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. Looking at the graph: it's a straight line passing through (0,2) and (4,10), which means the steepness (slope) is the same everywhere, so the rate of change is constant. Let's calculate the rate: From x = 0 to x = 4: Δy/Δx = (10 - 2)/(4 - 0) = 8/4 = 2. Since this is a straight line, this rate of 2 is constant everywhere on the line. Choice A correctly identifies the rate as constant at 2 because the calculation (10-2)/(4-0) = 8/4 = 2 gives the slope of the line. Choice D says a line does not have a constant rate of change, but this is completely backwards—lines are the ONLY graphs that have constant rates of change! That's what makes them linear! Visual shortcut: if the graph is a straight line, the rate is constant. Period. Any curvature—even slight—means non-constant rate. Straightness = constant rate, curvature = variable rate. This is the quickest check!
  5. Compare the rate of change of y=x2y=x^2 over two intervals: from x=1x=1 to x=2x=2 and from x=2x=2 to x=3x=3. Is the rate of change constant?
    Answer: No; average rates are 4−12−1=3\frac{4-1}{2-1}=3 and 9−43−2=5\frac{9-4}{3-2}=5, so it is not constant (nonlinear). 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. Let's check if the rate is constant by calculating Δy/Δx for the given intervals: From x = 1 to x = 2: y changes from 1² = 1 to 2² = 4, so Δy/Δx = (4 - 1)/(2 - 1) = 3/1 = 3. From x = 2 to x = 3: y changes from 2² = 4 to 3² = 9, so Δy/Δx = (9 - 4)/(3 - 2) = 5/1 = 5. The rates are different (3 and 5), so no, the rate is not constant. Choice B correctly calculates the average rates as 3 and 5, showing they are not constant, which confirms this is nonlinear. Choice D says the rate is constant because the second differences are constant, but this is backwards—constant second differences indicate a quadratic function, which has non-constant (first) rate of change. Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2x2^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.
  6. A cell phone plan charges a $25 monthly fee plus $0.10 per text message. After analyzing usage patterns, the company offers a new plan with a $35 monthly fee and $0.05 per text message. For what range of monthly text messages would the new plan result in a constant rate of savings compared to the original plan?
    Answer: The savings rate is constant at $0.05 per text message for all possible numbers of text messages sent monthly The difference between plans is: (25 + 0.10x) - (35 + 0.05x) = -10 + 0.05x. This shows the new plan saves $0.05 per text message consistently, regardless of volume. The rate of savings is constant for all values of x. A is wrong about the threshold. B incorrectly suggests the rate varies. D confuses the fixed difference with the rate of change.
  7. A gym charges a membership fee plus a fixed cost per visit. The total cost CC (in dollars) after vv visits is C=25+4vC = 25 + 4v. Is the rate of change of cost with respect to visits constant? If so, what is the rate?
    Answer: Yes; constant rate ΔC/Δv=4\Delta C/\Delta v = 4 dollars per visit. 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 2x2^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.
  8. Which situation represents a constant rate of change (and therefore can be modeled by a linear function)?
    Answer: A tank is filled at 3 liters per minute for 20 minutes. 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 tank is filled at 3 liters per minute for 20 minutes,' the language 'at 3 liters per minute' indicates constant rate. Each unit of input (minute) produces the same change in output (liters filled), specifically 3 liters per minute. Choice C correctly identifies the constant rate because the 'per minute' is fixed and steady, with no mention of changing speed. Choice A says the value decreases by 15% each year, but that's non-constant—percent change means exponential decay, where the amount lost changes each year; it's tempting if you miss the 'percent' clue, but 'percent' signals non-constant rate! 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! 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!
  9. Does the table show a constant rate of change between xx and yy? If so, what is the constant rate Δy/Δx\Delta y/\Delta x? Table (equal xx-intervals of 1): - xx: 0, 1, 2, 3, 4 - yy: 5, 8, 11, 14, 17
    Answer: Yes; constant rate Δy/Δx=3\Delta y/\Delta x = 3 (the yy-values increase by 3 each time xx increases by 1). 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!
  10. A water tank is being filled at a rate of 15 gallons per minute for the first 20 minutes, then at 8 gallons per minute for the next 30 minutes. Which statement best describes the rate of change of water volume with respect to time?
    Answer: The water volume changes at a constant rate during each interval, but the overall process does not have a constant rate 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.
  11. Determine whether the function f(x)=7x−5f(x)=7x-5 has a constant rate of change. If it does, what is the rate?
    Answer: Yes; constant rate =7=7 because the function is linear of the form y=mx+by=mx+b. 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 f(x)=7x-5: 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. Choice B correctly identifies the rate as constant at 7 because the function is linear of the form y=mx+b. Choice C says constant rate =-5 because -5 is the constant term, but the constant term is the y-intercept, not the rate; the rate is the coefficient of x in linear functions—don't confuse intercepts with slopes! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2x2^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.
  12. A swimming pool is being drained through two pipes. The water level decreases according to h(t)=48−6th(t) = 48 - 6t for the first 4 hours, then according to h(t)=24−2th(t) = 24 - 2t for the remaining time, where hh is height in inches and tt is time in hours from the start of each phase. What can be concluded about the rate of water level change?
    Answer: The water level decreases at constant rates during each phase: 6 inches/hour initially, then 2 inches/hour subsequently 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.
  13. A manufacturing machine produces widgets according to the function W(t)=45t+120W(t) = 45t + 120, where tt is time in hours and W(t)W(t) is the total number of widgets produced. A second machine follows W(t)=60t2−15t+50W(t) = 60t^2 - 15t + 50. Which statement correctly describes the rate of change for these machines?
    Answer: The first machine produces at a constant rate of 45 widgets per hour, while the second machine's rate varies with time 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.
  14. Determine whether the function f(x)=4x−7f(x)=4x-7 has a constant rate of change. If it does, what is the rate (the value of Δf/Δx\Delta f/\Delta x for any equal Δx\Delta x)?
    Answer: Yes; constant rate of change =4=4 because it is in the form y=mx+by=mx+b. 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 2x2^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.
  15. Which statement about constant rate of change is true for the function g(x)=7−4xg(x)=7-4x?
    Answer: The rate of change is constant and equals −4-4 because g(x)g(x) is linear of the form mx+bmx+b. 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 2x2^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.
  16. Does the table show a constant rate of change between xx and yy? Explain your choice by comparing Δy\Delta y for each 1-unit increase in xx. Table (equal xx-intervals of 1): - xx: 1, 2, 3, 4, 5 - yy: 2, 4, 8, 16, 32
    Answer: No; Δy\Delta y values are 2, 4, 8, 16, so the rate of change is not constant (nonlinear). 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 = 1 to x = 2: Δy/Δx = (4 - 2)/(2 - 1) = 2/1 = 2. From x = 2 to x = 3: Δy/Δx = (8 - 4)/(3 - 2) = 4/1 = 4. From x = 3 to x = 4: Δy/Δx = (16 - 8)/(4 - 3) = 8/1 = 8. From x = 4 to x = 5: Δy/Δx = (32 - 16)/(5 - 4) = 16/1 = 16. The rates are different (2,4,8,16), so no, the rate is not constant—it's changing. Choice B correctly identifies the rate as non-constant because the Δy values are 2,4,8,16, showing unequal differences that indicate a nonlinear (exponential) pattern. Choice A says yes with constant rate 2 because y doubles each time, but that's confusing constant ratio (multiplying by 2) with constant rate—constant ratio means exponential, not linear, so always check the differences, not just the ratios! 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! 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.
  17. Does the function f(x)=3x2−1f(x)=3x^2-1 have a constant rate of change? (A constant rate means Δy/Δx\Delta y/\Delta x is the same for all equal intervals.)
    Answer: No; it is nonlinear (quadratic), so Δy/Δx\Delta y/\Delta x is not constant. 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 2x2^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.
  18. Determine whether the function g(x)=2x2−3g(x)=2x^2-3 has a constant rate of change. Use the idea of Δy/Δx\Delta y/\Delta x over equal intervals (no calculus).
    Answer: No; it is nonlinear (quadratic), so the rate of change is not constant. 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 g(x) = 2x² - 3: This is a quadratic, and these function types have variable rates of change—the rate is different at different x-values, so it's not constant. Choice B correctly identifies it as non-constant because it is nonlinear (quadratic), meaning the rate varies as x changes. Choice A says yes with constant rate 2 because the coefficient of x² is 2, but that's confusing the leading coefficient with the rate—quadratics have rates that increase (or decrease) linearly, not stay constant; always check the degree of the polynomial! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2x2^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. Choice Y sees the formula [quadratic/exponential/etc.] and doesn't recognize that this function type has non-constant rate. Only linear functions (y = mx + b form) have constant rates. Quadratics, exponentials, and other nonlinear functions all have rates that vary as x changes!
  19. A taxi charges a flat fee of $4 plus $2 per mile. Does this situation involve a constant rate of change (change in cost per change in miles)?
    Answer: Yes; the rate is constant at 22 dollars per mile, so the relationship is linear. 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. In this context, 'a taxi charges a flat fee of $4 plus $2 per mile,' we analyze: the language '$2 per mile' indicates constant rate. Each additional mile produces the same change in cost, specifically $2 per mile. The flat fee of $4 doesn't change the rate—it just shifts the starting point up by $4 (the y-intercept). Choice B correctly identifies the rate as constant at $2 per mile because the cost increases by exactly $2 for each mile traveled, regardless of the initial $4 fee. Choice A says the rate changes because of the flat fee, but the flat fee is a one-time charge that doesn't affect the rate of change—the cost still increases by $2 for each mile, whether it's the 1st mile or the 100th mile. The foolproof test: if you can write the relationship as y = mx + b (here, Cost = 2(miles) + 4), then the rate is constant and equals m. The flat fee is just the b-value and doesn't affect the constant rate m!
  20. Which situations represent a constant rate of change? (1) A worker earns $18 per hour. (2) The area of a square is A=s2A=s^2 where ss is the side length. (3) A car's distance traveled increases by 60 miles each hour at a steady speed.
    Answer: Only (1) and (3) 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! Let's analyze each situation: (1) 'A worker earns $18 per hour' indicates constant rate—each hour produces the same change in earnings, specifically 18/hour. (2) 'The area of a square is A = s²' is a quadratic relationship, and quadratics have non-constant rates—as side length increases, the area increases faster and faster. (3) 'A car's distance traveled increases by 60 miles each hour at steady speed' indicates constant rate—the language 'steady speed' and 'each hour the same amount' confirms constant rate of 60 mph. Choice B correctly identifies only (1) and (3) as having constant rates because both describe situations where the change per unit time/input is fixed, while (2) involves a quadratic relationship with varying rate. Choice D incorrectly includes (2), not recognizing that A = s² represents a quadratic function where the rate of change of area with respect to side length is not constant (it equals 2s, which varies with s). 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!

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