Algebra 2 Quiz: Recognize Constant Rate Changes
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Recognize Constant Rate ChangesQuestion 1 of 20

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.

Yes; all polynomials have a constant rate of change.
Yes; the coefficient of xx is −6-6, so the constant rate of change is −6-6.
No; it is quadratic (not of the form mx+bmx+b), so the rate of change is non-constant.
No; because f(0)=1f(0)=1, the function cannot have a constant rate of change.
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Algebra 2 Quiz

Algebra 2 Quiz: Recognize Constant Rate Changes

Practice Recognize Constant Rate Changes in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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 2.

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.

All questions

Question 1

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.

  1. Yes; all polynomials have a constant rate of change.
  2. Yes; the coefficient of xx is −6-6, so the constant rate of change is −6-6.
  3. No; it is quadratic (not of the form mx+bmx+b), so the rate of change is non-constant. (correct answer)
  4. No; because f(0)=1f(0)=1, the function cannot have a constant rate of change.
Explanation: 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!

Question 2

The function f(x)=7x−3f(x)=7x-3 represents a relationship between xx and f(x)f(x). Is the rate of change constant? If yes, what is it?

  1. No; because there is a −3-3 in the formula, the rate of change is non-constant.
  2. Yes; the constant rate of change is 77 (linear function). (correct answer)
  3. Yes; the constant rate of change is −3-3 (linear function).
  4. No; because f(x)f(x) can be negative, the rate of change is non-constant.
Explanation: 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 = 7 for one interval and also 7 for every other equal interval, the rate is constant at 7. 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 function f(x) = 7x - 3 is already in the form y = mx + b, where m = 7 is the slope (constant rate of change) and b = -3 is the y-intercept. This form immediately tells us the function is linear with a constant rate of change equal to the coefficient of x, which is 7. To verify, we can check: if x increases by 1, then f(x) increases by 7(1) = 7; if x increases by 2, then f(x) increases by 7(2) = 14, which is exactly 2 times the change for a unit increase. Choice B correctly identifies that the constant rate of change is 7, recognizing this as a linear function. Choice C incorrectly claims the rate is -3, confusing the y-intercept (the constant term) with the rate of change (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. In the formula y = mx + b, m is the constant rate. In y = ab^x, b is the constant ratio. If neither form fits → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!

Question 3

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.

  1. Only (a) (correct answer)
  2. Only (b)
  3. Only (c)
  4. (a) and (c)
Explanation: 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!

Question 4

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?

  1. Both functions have a constant rate of change because both are increasing.
  2. Only Function 2 has a constant rate of change because it increases faster.
  3. Only Function 1 has a constant rate of change because its graph is a straight line (linear). (correct answer)
  4. Neither function has a constant rate of change because both pass through more than one point.
Explanation: 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!

Question 5

Two different relationships are shown in tables with equal xx-intervals.

Table 1: xx: 0, 1, 2, 3 yy: 4, 9, 14, 19

Table 2: xx: 0, 1, 2, 3 yy: 2, 4, 8, 16

Which table shows a constant rate of change (linear)?

  1. Only Table 1; its differences are 5, 5, 5 (constant rate). (correct answer)
  2. Only Table 2; its ratio y/xy/x is constant so the rate is constant.
  3. Both tables; any increasing table has a constant rate of change.
  4. Neither table; constant rate requires yy to start at 0 when x=0x=0.
Explanation: 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 Table 1, let's calculate Δy with Δx = 1: from x=0 to x=1, y goes from 4 to 9, so Δy = 5; from x=1 to x=2, y goes from 9 to 14, so Δy = 5; from x=2 to x=3, y goes from 14 to 19, so Δy = 5. Constant differences of 5 mean constant rate! For Table 2: from x=0 to x=1, y goes from 2 to 4, so Δy = 2; from x=1 to x=2, y goes from 4 to 8, so Δy = 4; from x=2 to x=3, y goes from 8 to 16, so Δy = 8. The differences (2, 4, 8) double each time—not constant! Note that Table 2 has constant ratios (each y is double the previous), indicating exponential growth. Choice A correctly identifies that only Table 1 has constant differences of 5, making it linear with constant rate. Choice B incorrectly focuses on constant ratio y/x for Table 2, but constant ratio of consecutive y-values characterizes exponential functions, not linear ones with 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. Table 1: differences are 5, 5, 5 → linear with rate 5. Table 2: ratios are 2, 2, 2 → exponential with base 2. If neither pattern is constant → some other type. Knowing which pattern to look for prevents confusing linear with exponential growth!

Question 6

A tank is filled at a steady rate. The volume of water VV (in liters) after tt minutes is shown.

tt: 0, 1, 2, 3, 4

VV: 10, 14, 18, 22, 26

Is the rate of change constant? If so, what is the constant rate ΔV/Δt\Delta V/\Delta t?

  1. Yes; constant rate of 44 liters per minute (linear). (correct answer)
  2. No; the volume increases, so the rate must be changing.
  3. Yes; constant rate of 1616 liters per minute because 26−10=1626-10=16.
  4. No; the differences are 4, 5, 4, 5 so it is non-constant.
Explanation: 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 = 4 for one interval and also 4 for every other equal interval, the rate is constant at 4. 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 calculate ΔV for each consecutive time interval: from t=0 to t=1, V goes from 10 to 14, so ΔV = 4; from t=1 to t=2, V goes from 14 to 18, so ΔV = 4; from t=2 to t=3, V goes from 18 to 22, so ΔV = 4; from t=3 to t=4, V goes from 22 to 26, so ΔV = 4. Since Δt = 1 for each step, we have ΔV/Δt = 4/1 = 4 liters per minute for every interval. Choice A correctly identifies that the rate is constant at 4 liters per minute, confirming this is a linear relationship. Choice D incorrectly claims the differences are 4, 5, 4, 5, but our calculations clearly show they're all 4—this error would lead to thinking the rate is non-constant when it actually is 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 4, 4, 4 → linear with rate 4. 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!

Question 7

A runner's distance dd (in meters) after tt seconds is shown.

tt: 0, 2, 4, 6, 8

dd: 0, 6, 12, 18, 24

Calculate Δd/Δt\Delta d/\Delta t for each interval and determine whether the rate of change is constant.

  1. Constant; each interval has Δd/Δt=3\Delta d/\Delta t = 3 meters per second, so it is linear. (correct answer)
  2. Non-constant; the differences in distance are 6, 6, 6, 6 so the rate changes.
  3. Constant; each interval has Δd/Δt=6\Delta d/\Delta t = 6 meters per second, so it is linear.
  4. Non-constant; because tt increases by 2 each time, you cannot check for constant rate.
Explanation: 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 = 3 for one interval and also 3 for every other equal interval, the rate is constant at 3. 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 calculate Δd/Δt for each interval, noting that Δt = 2 for each step: from t=0 to t=2, d goes from 0 to 6, so Δd = 6 and Δd/Δt = 6/2 = 3; from t=2 to t=4, d goes from 6 to 12, so Δd = 6 and Δd/Δt = 6/2 = 3; from t=4 to t=6, d goes from 12 to 18, so Δd = 6 and Δd/Δt = 6/2 = 3; from t=6 to t=8, d goes from 18 to 24, so Δd = 6 and Δd/Δt = 6/2 = 3. The rate is constant at 3 meters per second. Choice A correctly identifies that each interval has Δd/Δt = 3 meters per second, so the relationship is linear. Choice C incorrectly calculates the rate as 6, perhaps forgetting to divide by Δt = 2, confusing the change in distance (6 meters) with the rate of change (3 meters per second). 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. When x-intervals aren't 1, remember to divide: if Δy = 6 and Δx = 2, then rate = 6/2 = 3, not 6. If differences (after dividing) are 3, 3, 3 → linear with rate 3. If ratios are 2, 2, 2 → exponential with base 2. Knowing which pattern to look for prevents confusing linear with exponential growth!

Question 8

Points are plotted on a coordinate plane: (0,1)(0,1), (1,3)(1,3), (2,5)(2,5), (3,7)(3,7).

Does this graph represent a constant rate of change? (Assume the points are connected in order.)

  1. No; the graph is curved because the yy-values increase.
  2. Yes; the slope between consecutive points is 22, so the rate of change is constant and the graph is linear. (correct answer)
  3. Yes; because the ratio y/xy/x is constant, the rate of change is constant.
  4. No; the slope changes from 1 to 2 to 3, so the rate is not constant.
Explanation: 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 calculate the slope between consecutive points: from (0,1) to (1,3), slope = (3-1)/(1-0) = 2; from (1,3) to (2,5), slope = (5-3)/(2-1) = 2; from (2,5) to (3,7), slope = (7-5)/(3-2) = 2. All slopes equal 2—the rate is constant! These points lie on the line y = 2x + 1. Choice B correctly identifies that the slope between consecutive points is 2, confirming constant rate of change and a linear relationship. Choice D incorrectly claims the slope changes from 1 to 2 to 3 (those are actually the y-coordinates, not slopes!), and Choice C confuses constant ratio with constant rate—y/x values are undefined, 3, 2.5, 2.33, which vary and don't determine 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!

Question 9

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 asafunctionofthenumberoftextmessagesas a function of the number of text messages tt $?

  1. 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. (correct answer)
  2. C=0.32t+120C = 0.32t + 120, because the cost increases at a constant rate of $0.32 per message with a base fee of $120.
  3. C=6t+180C = 6t + 180, because the cost increases at a constant rate of $6 per message with a base fee of $180.
  4. C=38t+32C = 38t + 32, because the cost increases at a constant rate of $38 per message with a base fee of $32.
Explanation: 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.

Question 10

A company's revenue model shows that for every 50 additional customers gained, monthly revenue increases by $2,000. Currently, with 300 customers, the company earns $15,000 per month. Which statement best describes this situation?

  1. Revenue changes at a decreasing rate as more customers are added, creating a quadratic relationship between customer count and revenue.
  2. Revenue changes at a constant rate of $2,000 per additional customer, creating a linear relationship between customer count and revenue.
  3. Revenue changes at an increasing rate as more customers are added, creating an exponential relationship between customer count and revenue.
  4. Revenue changes at a constant rate of $40 per additional customer, creating a linear relationship between customer count and revenue. (correct answer)
Explanation: When you see a problem describing how one quantity changes in response to another, you need to identify the type of relationship and calculate the rate of change. Let's find the rate per customer. The problem states that 50 additional customers increase revenue by $2,000. To find the rate per individual customer, divide: $\frac{\2,000}{50 \text{ customers}} = $40 \text{ per customer} Since revenue increases by the same amount ($40) for each additional customer, this creates a linear relationship with a constant rate of change. Now let's examine why each answer choice is right or wrong: Choice A is incorrect because the rate of change is constant, not decreasing. A quadratic relationship would show acceleration or deceleration in the rate of change. Choice B makes a calculation error. It incorrectly states the rate as $2,000 per customer instead of recognizing that $2,000 applies to 50 customers, not one customer. Choice C is wrong because exponential relationships involve percentage-based growth rates that compound over time. Here, we have a fixed dollar amount added per customer, which is linear, not exponential. Choice D correctly identifies both the rate ($40 per customer) and the relationship type (linear). The constant rate of $40 per additional customer creates a straight-line relationship between customer count and revenue. Study tip: When analyzing rate problems, always check the units carefully. If the rate is "per group" (like per 50 customers), divide to find the rate per individual unit before determining the relationship type.

Question 11

A savings account balance is tracked over several months. In month 1, the balance is $800. In month 3, the balance is $920. In month 5, the balance is $1040. In month 7, the balance is $1160. Which statement accurately describes the rate of change in the account balance?

  1. The balance increases at a decelerating rate over time, indicating a logarithmic growth pattern in the savings account.
  2. The balance increases at a constant rate of $120 per month, indicating a linear growth pattern in the savings account.
  3. The balance increases at an accelerating rate over time, indicating an exponential growth pattern in the savings account.
  4. The balance increases at a constant rate of $60 per month, indicating a linear growth pattern in the savings account. (correct answer)
Explanation: When you see a problem tracking values over time, you need to determine the pattern of change by calculating the rate between data points. Let's examine the rate of change between consecutive data points. From month 1 to month 3 (2 months), the balance increases from $800 to $920, a change of $120. This gives us a rate of $1202=60\frac{120}{2} = 60 $ dollars per month. From month 3 to month 5 (2 months), the balance goes from $920 to $1040, another $120 increase over 2 months, again yielding $60 per month. From month 5 to month 7, we see $1040 to $1160, which is $120 over 2 months, maintaining the same $60 per month rate. Since the rate of change is constant at $60 per month, this represents linear growth, making answer D correct. Answer A is wrong because logarithmic growth shows a decelerating rate, but our rate stays constant. Answer B incorrectly calculates the rate as 120permonth—thisconfusesthetotalchangeover2−monthperiods(120 per month—this confuses the total change over 2-month periods (120) with the monthly rate ($60). Answer C suggests exponential growth with an accelerating rate, but again, our rate remains steady rather than increasing. Study tip: For rate of change problems, always divide the total change by the time interval to find the unit rate. If that rate stays the same between all intervals, you have linear growth. Watch out for trap answers that use the total change instead of the per-unit rate.

Question 12

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?

  1. The water is being added at a constant rate of 5 gallons per minute, indicating a linear relationship between time and total water volume. (correct answer)
  2. The water is being added at a constant rate of 15 gallons per minute, indicating a linear relationship between time and total water volume.
  3. The water is being added at an increasing rate over time, indicating an exponential relationship between time and total water volume.
  4. The water is being added at a decreasing rate over time, indicating a quadratic relationship between time and total water volume.
Explanation: 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.

Question 13

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?

  1. The elevator moves at a constant rate of 5 feet per second, demonstrating uniform motion with a linear height function.
  2. The elevator moves at a constant rate of 3 feet per second, demonstrating uniform motion with a linear height function. (correct answer)
  3. The elevator moves at an increasing rate over time, demonstrating accelerated motion with a quadratic height function.
  4. The elevator moves at a decreasing rate over time, demonstrating decelerated motion with a quadratic height function.
Explanation: 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.

Question 14

A factory produces widgets at different rates during various shifts. The production data shows: Morning shift produces 150 widgets in the first hour, 165 in the second hour, 180 in the third hour. Afternoon shift produces 200 widgets in the first hour, 225 in the second hour, 250 in the third hour. Evening shift produces 100 widgets in each of the first three hours.

Which shift demonstrates a constant rate of change in widget production per hour?

  1. The morning shift, because production increases by exactly 15 widgets each hour, showing a constant rate of change over time.
  2. The afternoon shift, because production increases by exactly 25 widgets each hour, showing a constant rate of change over time.
  3. The evening shift, because production remains at exactly 100 widgets each hour, showing a constant rate of change over time. (correct answer)
  4. All three shifts, because each demonstrates a predictable pattern of widget production that can be modeled mathematically over time.
Explanation: A constant rate of change means the difference between consecutive values remains the same. Evening shift: 100, 100, 100 (differences of 0, 0). This represents a constant rate of change of 0 widgets per hour. Choice A shows increasing rates (15 widgets per hour increase). Choice B shows increasing rates (25 widgets per hour increase). Choice D is incorrect because having a predictable pattern doesn't mean the rate of change is constant.

Question 15

Two relationships are shown below.

Relationship 1: x: 0, 1, 2, 3; y: 7, 9, 11, 13 Relationship 2: x: 0, 1, 2, 3; y: 2, 4, 8, 16

Which statement is correct about constant rate of change?

  1. Only Relationship 2 has constant rate of change because it has a constant ratio.
  2. Neither has constant rate of change because neither starts at y=0y=0.
  3. Only Relationship 1 has constant rate of change because Δy\Delta y is constant for equal Δx\Delta x (linear). (correct answer)
  4. Both relationships have constant rate of change because both y-values increase.
Explanation: 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 Relationship 1: The differences are 9-7=2, 11-9=2, 13-11=2, giving constant Δy/Δx = 2/1 = 2 for all intervals—this is linear with constant rate! For Relationship 2: The differences are 4-2=2, 8-4=4, 16-8=8, giving rates of 2, 4, 8—these double each time, indicating exponential growth where y = 2^(x+1). The ratios are constant (4/2=2, 8/4=2, 16/8=2), but the rates vary. Choice C correctly identifies that only Relationship 1 has constant rate of change because its differences are constant, making it linear. Choice B confuses constant ratio (which characterizes exponential functions) with constant rate (which characterizes linear functions). 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!

Question 16

A phone plan charges a $15 monthly fee plus $0.10 per text message. Let CC be the total cost (in dollars) for xx text messages.

Is the rate of change of CC with respect to xx constant? If so, what is it and what type of function models the situation?

  1. No; because there is a monthly fee, the rate of change is not constant.
  2. Yes; constant rate of 0.100.10 dollars per text, so the model is linear. (correct answer)
  3. Yes; constant rate of 1515 dollars per text, so the model is linear.
  4. No; because the cost increases, the rate must increase too.
Explanation: 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 model C=15+0.10x is linear, so constant rate 0.10. Choice B correctly identifies the constant rate of 0.10 because it's y=mx+b form, modeling linear cost. Choice A wrongly assumes a fixed fee means non-constant rate, but the fee is the b (constant), not affecting the steady per-text rate m! 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!

Question 17

A runner's distance dd (in meters) after tt seconds is shown.

tt (s)0246
dd (m)0102030

Is the rate of change constant? If so, what is the constant rate Δd/Δt\Delta d/\Delta t?

  1. Yes; constant rate of 55 m/s because Δd/Δt=10/2=5\Delta d/\Delta t = 10/2 = 5 for each interval. (correct answer)
  2. No; because time increases by 2 seconds each step, the rate cannot be constant.
  3. Yes; constant rate of 1010 m/s because distance increases by 10 each row.
  4. No; because the ratio d/td/t changes, the rate of change is not constant.
Explanation: 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! Δd for Δt=2 are 10,10,10, all equal, so Δd/Δt=5 constantly. Choice A correctly identifies the constant rate of 5 because the ratios are equal over equal intervals. Choice C mistakes Δd=10 for the rate, but divide by Δt=2 to get the actual rate 5—always compute Δy/Δx! 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!

Question 18

A tank is being filled. The volume of water VV (in liters) is recorded every minute tt.

tt (min)01234
VV (L)1013172228

Calculate the rate of change over each 1-minute interval. Is the rate of change constant?

  1. No; ΔV\Delta V values (3, 4, 5, 6) are not all equal, so the rate is non-constant (nonlinear). (correct answer)
  2. Yes; the increases are 3, 4, 5, 6 so the constant rate is 4.5 L/min.
  3. Yes; because time increases by 1 each row, the rate must be constant.
  4. No; but only the first interval matters, and it is 3 L/min so the rate is constant at 3 L/min.
Explanation: 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! Here, ΔV for Δt=1 are 3,4,5,6, which vary, so non-constant rate. Choice B correctly identifies the non-constant rate because the ΔV values are not equal, indicating nonlinear. A mistake like in Choice A is averaging varying differences, but for constant rate, they must be identical, not averaged! 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!

Question 19

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?

  1. Plan 1 only; it increases by a constant $5 per month (linear). (correct answer)
  2. Plan 2 only; it has a constant rate because it multiplies by 2 each month.
  3. Both plans; any repeating pattern is a constant rate of change.
  4. Neither plan; any sign-up fee makes the rate of change non-constant.
Explanation: 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!

Question 20

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 smoothly curved upward (not a straight line).

  1. Yes; because the points increase, the rate of change is constant.
  2. No; the graph is curved, so the slope changes and the rate of change is non-constant (nonlinear). (correct answer)
  3. Yes; because Δy\Delta y values are 1,3,51, 3, 5, which are all odd numbers.
  4. Yes; because the ratio yx\frac{y}{x} is constant across the points.
Explanation: 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: from (0,0) to (1,1), rate=1/1=1; (1,1) to (2,4), Δy=3, Δx=1, rate=3; (2,4) to (3,9), Δy=5, Δx=1, rate=5—increasing, non-constant. Choice B correctly identifies the non-constant rate because the curve means changing slopes, indicating nonlinear (like quadratic). Choice D might check ratios y/x (0/0 undefined,1/1=1,4/2=2,9/3=3—not constant), but that's for exponentials—use slopes or linearity for rate constancy instead! 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!