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Recognize Constant Rate Changes - Algebra 2

  1. A graph on a coordinate plane shows a curve (not a straight line) that passes through (0,0)(0,0), (1,1)(1,1), and (2,4)(2,4). Does this graph represent a constant rate of change? Choose the best answer.
    Answer: No; the graph is curved, so the slope (rate of change) is not constant, meaning it is nonlinear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! The graph is curved, with slopes like (1-0)/(1-0)=1 but (4-1)/(2-1)=3, varying, so non-constant. Choice C correctly identifies the non-constant rate because it's curved, meaning nonlinear. Choice B miscalculates one slope and assumes it's constant, but check multiple intervals—curves have changing slopes! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  2. Which situation has a constant rate of change (and therefore can be modeled by a linear function)? (a) A tank fills 4 gallons every minute. (b) The area of a square as its side length increases. (c) A bacteria population doubles every hour.
    Answer: Only (a) This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Situation (a) fills 4 gallons per minute, so constant Δgallons/Δtime=4, linear; (b) area A=s², rate dA/ds=2s varies with s, quadratic; (c) doubles hourly, constant ratio 2 but varying differences (e.g., 1 to 2, +1; 2 to 4, +2), exponential. Choice A correctly identifies only (a) as having constant rate because it's additive growth, equivalent to linear. For (c), a common error is thinking doubling is constant rate, but that's constant ratio—calculate differences to confirm they increase, not stay equal! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  3. A runner's total distance dd (in miles) after tt hours is given by d=6t+1d=6t+1. Is the rate of change constant? If so, what is the constant rate and what type of function is it?
    Answer: Yes; constant rate 66 miles per hour, and it is linear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! The formula d=6t+1 is in mx+b form with m=6, so constant rate of 6; for example, from t=0 to 1, Δd=(6*1+1)-(1)=7-1=6; t=1 to 2, (12+1)- (6+1)=13-7=6—constant. Choice A correctly identifies the constant rate of 6 and linear type because it's mx+b form. Choice B incorrectly blames the +1 for non-constancy, but constants affect intercept, not rate—focus on the coefficient of t for the rate! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  4. Consider the function f(x)=4x−7f(x) = 4x - 7. Is the rate of change constant? If so, what is the constant rate Δf/Δx\Delta f/\Delta x?
    Answer: Yes; constant rate of 44, so it is linear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! The formula f(x)=4x-7 is in y=mx+b form, so constant rate m=4. Choice A correctly identifies the constant rate of 4 because it's linear, and negatives don't affect rate constancy. A distractor like Choice C might confuse the y-intercept -7 with the rate, but rate is the coefficient of x! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  5. Function 1 is given by f(x)=5x−1f(x)=5x-1. Function 2 is given by g(x)=x2−1g(x)=x^2-1. Which statement correctly compares their rates of change?​​​
    Answer: Only f(x)f(x) has a constant rate of change (slope 5), so ff is linear and gg is nonlinear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Function f(x) = 5x - 1 is in the form y = mx + b with m = 5, so it has constant rate of change 5. Function g(x) = x² - 1 is quadratic; let's check its rate: from x=0 to x=1, Δg/Δx = (0-(-1))/1 = 1; from x=1 to x=2, Δg/Δx = (3-0)/1 = 3; from x=2 to x=3, Δg/Δx = (8-3)/1 = 5. The rates 1, 3, 5 are increasing—not constant! Choice D correctly states that only f(x) has a constant rate of change (slope 5), making f linear and g nonlinear. Choice A incorrectly claims both have constant rates just because they're polynomials (only degree-1 polynomials are linear!), and Choice C reverses the truth by claiming the quadratic has constant rate. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  6. Does the table show a constant rate of change between xx and yy? Explain using Δy/Δx\Delta y/\Delta x. Table (equal xx-intervals of 1): xx: 0, 1, 2, 3 yy: 4, 7, 10, 13
    Answer: Yes; Δy/Δx=3\Delta y/\Delta x = 3 for each interval, so the rate is constant and the relationship is linear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! For this table, calculate the rates: from x=0 to 1, Δy/Δx = (7-4)/1 = 3; from x=1 to 2, (10-7)/1 = 3; from x=2 to 3, (13-10)/1 = 3, so the rate is constant at 3. Choice B correctly identifies the constant rate because the differences are equal, indicating a linear relationship. A common distractor like choice A might miscalculate the differences as varying, but actually they are consistently 3—double-check your subtractions to avoid this error. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  7. A water tank is being filled. The amount of water WW (in liters) after tt minutes is recorded. Is the rate of change ΔW/Δt\Delta W/\Delta t constant? Use the table to decide. t: 0, 2, 4, 6 W: 10, 18, 26, 34
    Answer: Yes; ΔW/Δt=4\Delta W/\Delta t = 4 liters per minute for each interval, so it is constant and linear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Notice the time intervals are equal (Δt = 2 minutes each). Let's calculate ΔW/Δt: from t=0 to t=2, ΔW/Δt = (18-10)/(2-0) = 8/2 = 4 liters/minute; from t=2 to t=4, ΔW/Δt = (26-18)/(4-2) = 8/2 = 4 liters/minute; from t=4 to t=6, ΔW/Δt = (34-26)/(6-4) = 8/2 = 4 liters/minute. All rates equal 4 liters per minute—constant! Choice A correctly identifies the constant rate of 4 liters per minute and recognizes this as linear. Choice C makes the error of looking at total change (8 liters) over each 2-minute interval instead of the rate per minute, while Choice B incorrectly claims the rate isn't constant. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  8. Does the function f(x)=3x2−6x+1f(x)=3x^2-6x+1 have a constant rate of change? Explain using the idea of linear vs. nonlinear functions.
    Answer: No; it is quadratic (not of the form mx+bmx+b), so the rate of change is non-constant. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! For f(x)=3x²-6x+1, it's quadratic (degree 2, not 1), so the rate varies; for example, from x=0 to 1, Δf=(3-6+1)-(1)= -2 to -1? Wait, f(0)=1, f(1)=3-6+1=-2, Δf=-3; f(2)=12-12+1=1, Δf from 1 to 2=3—not constant. Choice B correctly identifies it as quadratic and thus non-constant rate because it's not in mx+b form. A distractor like choice A might misread the -6x term as the constant rate, but ignore the x² which makes the rate change—always check the highest degree for linearity! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  9. Two tables describe relationships between xx and yy at equal xx-intervals. Table 1: x: 0, 1, 2, 3; y: 6, 9, 12, 15 Table 2: x: 0, 1, 2, 3; y: 2, 4, 8, 16 Which statement is correct?
    Answer: Only Table 1 shows constant rate of change because its first differences are constant, so it is linear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Table 1 has Δy=3,3,3 constant; Table 2 has ratios 2,2,2 constant but differences 2,4,8 varying. Choice C correctly identifies only Table 1 has constant rate because its first differences are equal, making it linear. Choice B confuses constant ratio (exponential) with constant rate—Table 2 is exponential, not linear! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  10. A coordinate plane shows two functions: - Function 1 is a straight line passing through points (0,1)(0,1) and (4,9)(4,9). - Function 2 is a curved graph passing through points (0,1)(0,1), (2,5)(2,5), and (4,17)(4,17). Which statement is true about constant rate of change?
    Answer: Only Function 1 has a constant rate of change because its graph is a straight line (linear). This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! For these graphs, Function 1 is straight, so constant rate; Function 2 is curved with varying slopes like (5-1)/(2-0)=2 but (17-5)/(4-2)=6, not constant. Choice C correctly identifies only Function 1 has constant rate because its graph is a straight line, making it linear. A distractor like Choice A might assume any increasing function has constant rate, but that's not true—curves can increase with changing rates! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  11. From the graph, determine if the rate of change is constant. The curve passes through (0,0)(0,0), (1,1)(1,1), (2,4)(2,4), and (3,9)(3,9) and is smooth and curved upward (not a straight line).
    Answer: No; the graph is curved, so the slope changes and the rate of change is non-constant (nonlinear). This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! The graph is curved upward through (0,0), (1,1), (2,4), (3,9), so slopes vary: e.g., from (0,0) to (1,1) rate=1, from (1,1) to (2,4) rate=3, from (2,4) to (3,9) rate=5, non-constant. Choice B correctly identifies the non-constant rate because the curve means changing slope, indicating nonlinear. A distractor like choice A might assume any increasing graph has constant rate, but curvature shows varying rates—look for straightness to confirm constancy. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  12. An elevator's height above ground level is recorded as it moves between floors. The data shows: at 10 seconds the elevator is at 25 feet, at 15 seconds it's at 40 feet, at 20 seconds it's at 55 feet, and at 25 seconds it's at 70 feet. What can be concluded about the elevator's motion?
    Answer: The elevator moves at a constant rate of 3 feet per second, demonstrating uniform motion with a linear height function. Calculate the rate of change: (40-25)/(15-10) = 15/5 = 3 feet per second. Checking other intervals: (55-40)/(20-15) = 15/5 = 3 feet per second, and (70-55)/(25-20) = 15/5 = 3 feet per second. The constant rate of 3 feet per second indicates uniform motion. Choice A miscalculates the rate as 5 feet per second. Choices C and D suggest non-uniform motion, but the constant rate contradicts these options.
  13. A tank is being filled so that the volume increases by 12 liters every 3 minutes. Is the rate of change of volume with respect to time constant? If so, what is the constant rate in liters per minute?
    Answer: Yes; constant rate =4=4 liters per minute, so the relationship is linear. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! The volume increases by 12 liters every 3 minutes, so rate = 12/3 = 4 liters per minute constantly, as it's a steady fill implying linear: V(t) = 4t (assuming starts at 0). Choice A correctly identifies the constant rate because the increase is proportional with fixed rate 4, a linear relationship. A distractor like choice D might see the ratio 12:3 and think it's constant ratio, but that's actually the rate (difference over interval)—distinguish by noting ratios would be for multiplicative growth, not additive here. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  14. A student is analyzing data from an experiment. The table shows measurements taken at regular intervals. Which conclusion about the rate of change is most accurate?
    Answer: The data shows a constant rate of change of 4 units per interval, indicating a linear relationship between time and measurement. Looking at consecutive differences: 12-8=4, 16-12=4, 20-16=4, 24-20=4. The constant difference of 4 units per interval indicates a linear relationship with constant rate of change. Choice B incorrectly calculates the rate as 8. Choice C suggests exponential growth, which would show increasing rates between consecutive terms. Choice D suggests quadratic behavior with decreasing rates, which doesn't match the constant differences observed.
  15. Which situation represents a constant rate of change (a linear relationship)?
    Answer: A worker earns $18 per hour for hh hours (no overtime). This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 18 for one interval and also 18 for every other equal interval, the rate is constant at 18. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Let's analyze each situation: (A) 15% decrease each year means multiplying by 0.85 each year—this is exponential decay with constant ratio, not constant rate; (B) Area = πr², a quadratic function in r, so rate increases as r increases; (C) Earnings = 18h, which is y = mx form with m = 18, a linear function with constant rate $18/hour; (D) Distance fallen follows d = ½gt², quadratic in time, with increasing rate of change. Choice C correctly represents a constant rate of change because the worker earns exactly $18 for each additional hour worked—a perfect linear relationship. Choice A represents exponential decay where the amount of decrease changes each year (15% of a smaller value each time), not a constant rate of change. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. "15% decrease" signals ratio (multiply by 0.85), not rate. "$18 per hour" signals rate (add $18). "Area of circle" involves squaring (quadratic). "Acceleration" means changing rate. Knowing these keywords prevents confusing linear with other function types!
  16. A cell phone plan charges a monthly base fee plus a fixed rate per text message. In January, Sarah sent 120 text messages and paid $32. In February, she sent 180 text messages and paid $38. Which expression represents the total monthly cost CC as a function of the number of text messages tt?
    Answer: C=0.10t+20C = 0.10t + 20, because the cost increases at a constant rate of $0.10 per message with a base fee of $20. The change in cost is $38 - $32 = $6 for 180 - 120 = 60 additional messages, giving a rate of $6 ÷ 60 = $0.10 per message. Using point-slope form with (120, 32): C = 0.10t + b. Substituting: 32 = 0.10(120) + b, so b = 20. Therefore C = 0.10t + 20. Choice B uses incorrect values from the problem setup. Choice C miscalculates the rate as $6 instead of $0.10. Choice D confuses the given data points with the rate and intercept.
  17. A water tank is being filled at a steady rate. After 3 minutes, the tank contains 45 gallons. After 8 minutes, the tank contains 70 gallons. If this pattern continues, which statement best describes the relationship between time and the total amount of water in the tank?
    Answer: The water is being added at a constant rate of 5 gallons per minute, indicating a linear relationship between time and total water volume. The change in water volume is 70 - 45 = 25 gallons over 8 - 3 = 5 minutes, giving a rate of 5 gallons per minute. Since this rate is constant, the relationship is linear. Choice B incorrectly calculates the rate as 15 gallons per minute. Choice C suggests exponential growth, which would show an increasing rate of change. Choice D suggests quadratic behavior with a decreasing rate, which doesn't match the constant rate observed.
  18. Two payment plans are described below. Plan 1: You pay a $10 sign-up fee plus $5 per month. Plan 2: You pay $5 the first month, $10 the second month, $20 the third month, and $40 the fourth month. Which plan has a constant rate of change in total cost with respect to months?
    Answer: Plan 1 only; it increases by a constant $5 per month (linear). This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Plan 1: total =10+5m, linear with constant Δcost/Δmonth=5; Plan 2: 5,10,20,40..., differences 5,10,20 varying (ratios=2 constant, exponential). Choice A correctly identifies only Plan 1 as constant rate because of equal additions, linear growth. Choice B mistakes Plan 2's constant ratio (multiplying by 2) for constant rate—remember to check differences for linear, ratios for exponential, to distinguish them! The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  19. Which situation represents a constant rate of change (and therefore a linear relationship)?​​​
    Answer: A tank is filled at 4 gallons per minute. This question tests your ability to recognize when a relationship has constant rate of change—the defining characteristic of linear functions. Constant rate of change means that for every unit increase in x, y changes by the same amount every time: if Δy/Δx = 5 for one interval and also 5 for every other equal interval, the rate is constant at 5. This constant rate is exactly what makes a function linear (y = mx + b where m is that constant rate). Only linear functions have this property—quadratics, exponentials, and other nonlinear functions have rates that vary at different x-values! Let's analyze each situation: (A) Area = s², so if s goes 1, 2, 3, then A goes 1, 4, 9 with differences 3, 5—not constant! (B) 5% growth means multiply by 1.05 each month—that's exponential with constant ratio, not constant rate. (C) 4 gallons per minute means volume = 4t, which is linear with constant rate 4. (D) Acceleration means changing velocity—the rate of position change is itself changing! Choice C correctly identifies a constant rate of change because adding 4 gallons every minute creates equal differences: after 1 min: 4 gal, after 2 min: 8 gal, after 3 min: 12 gal, with constant differences of 4. Choice A fails because quadratic growth (s²) has increasing rates, and Choice B represents exponential growth with constant ratio, not constant rate. The three-method constant rate test: METHOD 1 (from table): Calculate Δy/Δx for each pair of consecutive points with equal Δx. All equal? Constant rate. Vary? Non-constant. METHOD 2 (from graph): Is it a straight line? Yes = constant rate. Curved? Non-constant. METHOD 3 (from formula): Is it y = mx + b form? Yes = constant rate m. Any other form (x², bxb^x, etc.)? Non-constant. Pick the method matching your representation! Don't confuse constant RATE with constant RATIO: constant rate (Δy/Δx equal) characterizes linear functions, constant ratio (y₂/y₁ equal) characterizes exponential functions. Check BOTH in a table: if differences are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. If neither constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!
  20. Examine the graph showing temperature changes throughout the day. During which time interval does the temperature change at a constant rate?
    Answer: From 2 PM to 6 PM, where temperature falls at a steady rate of 2°F per hour during the afternoon cooling period. A constant rate of change appears as a straight line segment on the graph. From 2 PM to 6 PM, the temperature decreases linearly from 80°F to 72°F over 4 hours, giving a constant rate of (72-80)/(6-2) = -8/4 = -2°F per hour. Choice A shows a curved warming period with changing rates. Choice B shows a curved segment during peak heating. Choice D shows a curved cooling segment with varying rates.

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