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Algebra

Algebra Help: Recognize Constant Rate Changes

Review real example questions for Recognize Constant Rate Changes in Algebra.

Question 1

Does the table show a constant rate of change between $x$ and $y$ ? If so, what is the constant rate $\Delta y/\Delta x$ ?

Table (equal $x$ -intervals of 1):

$x$ : 0, 1, 2, 3, 4
$y$ : 5, 8, 11, 14, 17

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

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

Question 2

A taxi charges a flat fee of $\$ 4 $plus$ $2 $for each mile traveled. Does this situation involve a constant rate of change, and if so what is the rate (in$ $/mile)?

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

Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. In real-world contexts, constant rate sounds like: 'travels at steady 60 mph,' 'costs

5 per item,' 'fills at 10 gallons per minute'—the 'per' language and steady/constant/fixed words signal constant rate. Non-constant rate sounds like: 'accelerating,' 'slowing down,' 'doubling each hour,' 'speed increasing'—these signal that the rate itself is changing! In this context, 'a taxi charges a flat fee of

4 plus

2 for each mile traveled,' we analyze: the language 'plus

2 for each mile' indicates constant rate. Each mile produces the same change in cost, specifically

2 per mile. Choice A correctly identifies the rate as constant at 2 because the cost increases by

2 for each additional mile, showing the linear nature of the relationship. Choice B says the flat fee means the rate cannot be constant, but the flat fee is just the y-intercept in a linear function, which doesn't affect the constancy of the rate—the rate is still constant as long as the per-mile charge is fixed. It's easy to think intercepts disrupt constancy, but they don't; only changes in the per-unit amount would make it non-constant! Context language decoder: words like 'constant speed,' 'steady rate,' '$X per unit,' 'every hour the same amount' → constant rate (linear). Words like 'accelerating,' 'percent per year,' 'doubling,' 'slowing down,' 'squared' → non-constant rate (nonlinear). The language almost always reveals which type!

Question 3

A plant’s height $h$ (in cm) is measured each week $w$ . The data are shown below.

Does the table show a constant rate of change of height with respect to time? If not, choose the statement that best describes why.

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

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

Question 4

A savings account balance $B$ (in dollars) is recorded at the end of each week $w$ .

Does the table represent a constant rate of change of $B$ with respect to $w$ ? Choose the best answer.

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

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

Question 5

Does the table show a constant rate of change between $t$ (time in hours) and $d$ (distance in miles)? If so, what is the constant rate $\Delta d/\Delta t$ ?

Table (equal $t$ -intervals of 2 hours):

$t$ : 0, 2, 4, 6
$d$ : 0, 120, 240, 360

Yes; constant rate $\Delta d/\Delta t = 360$ miles per hour.
Yes; constant rate $\Delta d/\Delta t = 60$ miles per hour.
No; the rate is not constant because the distances are different each time.
Yes; constant rate $\Delta d/\Delta t = 120$ miles per hour.

\Delta y / \Delta x

(change in y over change in x) for each consecutive pair of points. If you get the same number every time, the rate is constant. If the values differ, the rate is non-constant. Example: if differences are 3, 3, 3, 3—constant! If differences are 2, 4, 6, 8—non-constant (actually quadratic pattern). Let's check if the rate is constant by calculating

\Delta y / \Delta x

for each interval in the table: From

t = 0

t = 2

\Delta d / \Delta t = (120 - 0)/(2 - 0) = 120/2 = 60

. From

t = 2

t = 4

\Delta d / \Delta t = (240 - 120)/(4 - 2) = 120/2 = 60

. From

t = 4

t = 6

\Delta d / \Delta t = (360 - 240)/(6 - 4) = 120/2 = 60

. All rates equal 60, so yes, constant rate of 60! Choice C correctly identifies the rate as constant with

\Delta d / \Delta t = 60

because after dividing the equal

\Delta d

(120) by

\Delta t

(2), we get consistent 60 mph across intervals. Choice B says yes with 120, but that's forgetting to divide by

\Delta t=2

—it's easy to just look at

\Delta d

without the 'per hour' part; always compute the full ratio

\Delta y / \Delta x

! The foolproof test for constant rate from a table: (1) Make sure your x-values increase by the same amount each time (like going 1, 2, 3, 4 or 0, 5, 10, 15), (2) Calculate the differences in y-values: y₂ - y₁, y₃ - y₂, y₄ - y₃, etc., (3) If all differences are equal, rate is constant! If they differ, rate is not constant. This works every time with equally-spaced x-values. If x-intervals aren't equal, you must divide each

\Delta y

by its

\Delta x

to check if the ratios are equal—that's key for non-uniform spacing!

Question 6

A gym charges a membership fee plus a fixed cost per visit. The total cost $C$ (in dollars) after $v$ visits is $C = 25 + 4v$ . Is the rate of change of cost with respect to visits constant? If so, what is the rate?

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

5 per item,' 'fills at 10 gallons per minute'—the 'per' language and steady/constant/fixed words signal constant rate. Non-constant rate sounds like: 'accelerating,' 'slowing down,' 'doubling each hour,' 'speed increasing'—these signal that the rate itself is changing! In this context, 'a gym charges a membership fee plus a fixed cost per visit with C = 25 + 4v,' we analyze: the language 'fixed cost per visit' indicates constant rate. Each unit of input (visit) produces the same change in output (cost), specifically 4 dollars per visit. Choice A correctly identifies the rate as constant with ΔC/Δv = 4 because the 'per visit' term is fixed, and the membership fee is just a starting point that doesn't affect the rate of change. Choice B says no because of the

25 membership fee, but that's a supportive reminder—the fee is like the y-intercept in y=mx+b, which doesn't change the constant slope m; it's easy to think constants make it nonlinear, but they don't! Context language decoder: words like 'constant speed,' 'steady rate,' '$X per unit,' 'every hour the same amount' → constant rate (linear). Words like 'accelerating,' 'percent per year,' 'doubling,' 'slowing down,' 'squared' → non-constant rate (nonlinear). The language almost always reveals which type! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.

Question 7

Determine whether the function $g(x) = x^2 + 2x$ has a constant rate of change.

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

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

Question 8

A savings account balance is modeled by $B(t)=100\cdot(1.5)^t$ , where $t$ is the number of months. Is the rate of change $\Delta B/\Delta t$ constant? (No calculus.)

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

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

Question 9

Does the function $f(x)=3x^2-1$ have a constant rate of change? (A constant rate means $\Delta y/\Delta x$ is the same for all equal intervals.)

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

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

Question 10

Determine whether the function $g(x)=2x^2+1$ has a constant rate of change.

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

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

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Algebra

Algebra Help: Recognize Constant Rate Changes

Review real example questions for Recognize Constant Rate Changes in Algebra.

Question 1

Does the table show a constant rate of change between $x$ and $y$ ? If so, what is the constant rate $\Delta y/\Delta x$ ?

Table (equal $x$ -intervals of 1):

$x$ : 0, 1, 2, 3, 4
$y$ : 5, 8, 11, 14, 17

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

Question 2

A taxi charges a flat fee of $\$ 4 $plus$ $2 $for each mile traveled. Does this situation involve a constant rate of change, and if so what is the rate (in$ $/mile)?

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

5 per item,' 'fills at 10 gallons per minute'—the 'per' language and steady/constant/fixed words signal constant rate. Non-constant rate sounds like: 'accelerating,' 'slowing down,' 'doubling each hour,' 'speed increasing'—these signal that the rate itself is changing! In this context, 'a taxi charges a flat fee of

4 plus

2 for each mile traveled,' we analyze: the language 'plus

2 for each mile' indicates constant rate. Each mile produces the same change in cost, specifically

2 per mile. Choice A correctly identifies the rate as constant at 2 because the cost increases by

Question 3

A plant’s height $h$ (in cm) is measured each week $w$ . The data are shown below.

Does the table show a constant rate of change of height with respect to time? If not, choose the statement that best describes why.

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

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

Question 4

A savings account balance $B$ (in dollars) is recorded at the end of each week $w$ .

Does the table represent a constant rate of change of $B$ with respect to $w$ ? Choose the best answer.

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

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

Question 5

Does the table show a constant rate of change between $t$ (time in hours) and $d$ (distance in miles)? If so, what is the constant rate $\Delta d/\Delta t$ ?

Table (equal $t$ -intervals of 2 hours):

$t$ : 0, 2, 4, 6
$d$ : 0, 120, 240, 360

Yes; constant rate $\Delta d/\Delta t = 360$ miles per hour.
Yes; constant rate $\Delta d/\Delta t = 60$ miles per hour.
No; the rate is not constant because the distances are different each time.
Yes; constant rate $\Delta d/\Delta t = 120$ miles per hour.

\Delta y / \Delta x

\Delta y / \Delta x

for each interval in the table: From

t = 0

t = 2

\Delta d / \Delta t = (120 - 0)/(2 - 0) = 120/2 = 60

. From

t = 2

t = 4

\Delta d / \Delta t = (240 - 120)/(4 - 2) = 120/2 = 60

. From

t = 4

t = 6

\Delta d / \Delta t = (360 - 240)/(6 - 4) = 120/2 = 60

. All rates equal 60, so yes, constant rate of 60! Choice C correctly identifies the rate as constant with

\Delta d / \Delta t = 60

because after dividing the equal

\Delta d

(120) by

\Delta t

(2), we get consistent 60 mph across intervals. Choice B says yes with 120, but that's forgetting to divide by

\Delta t=2

—it's easy to just look at

\Delta d

without the 'per hour' part; always compute the full ratio

\Delta y / \Delta x

\Delta y

by its

\Delta x

to check if the ratios are equal—that's key for non-uniform spacing!

Question 6

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

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

Question 7

Determine whether the function $g(x) = x^2 + 2x$ has a constant rate of change.

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

Question 8

A savings account balance is modeled by $B(t)=100\cdot(1.5)^t$ , where $t$ is the number of months. Is the rate of change $\Delta B/\Delta t$ constant? (No calculus.)

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

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

Question 9

Does the function $f(x)=3x^2-1$ have a constant rate of change? (A constant rate means $\Delta y/\Delta x$ is the same for all equal intervals.)

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

Question 10

Determine whether the function $g(x)=2x^2+1$ has a constant rate of change.

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

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