What is Slope & rate of change on the Algebra 1 Regents?

Slope & rate of change is a Linear Functions topic on the Algebra 1 Regents. RegentsLab has 21 practice questions on it, each with a full worked solution.

How do you study Slope & rate of change for the Algebra 1 Regents?

Work a range of Slope & rate of change question types and read the worked solution every time you miss one — the exam reuses the same ideas year after year. Start with the questions on this page.

Slope & rate of change: Algebra 1 Regents Practice

Practice Slope & rate of change for the Algebra 1 Regents with real-style questions and full worked solutions. Miss one? Read the solution — the exam reuses the same ideas every year.

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Question 8 · 2 pts

What is the slope of the line through $(1,\,2)$ and $(4,\,11)$ ?

(A) $3$ ✓
(B) $9$
(C) $\tfrac{1}{3}$
(D) $-3$

Worked solution

$m = \frac{11-2}{4-1} = \frac{9}{3}$ Rise over run.

$= 3$

Answer: m = 3

Slope is the change in y over the change in x between two points.

Question 13 · 2 pts

What is the slope of the line that passes through the points $(1, 2)$ and $(4, 11)$?

(A) $9$
(B) $3$ ✓
(C) $-3$
(D) $\frac{1}{3}$

Worked solution

$m = \frac{y_2 - y_1}{x_2 - x_1}$ Slope formula

$m = \frac{11 - 2}{4 - 1}$ Substitute the two points

$m = \frac{9}{3} = 3$ Simplify

Answer: $3$

Slope measures how much y changes per unit change in x; apply the slope formula (change in y divided by change in x) to any two points on the line.

Question 14 · 2 pts

The table below shows a linear relationship between $x$ and $y$. | $x$ | 0 | 2 | 4 | 6 | |---|---|---|---|---| | $y$ | 5 | 11 | 17 | 23 | What is the rate of change of $y$ with respect to $x$?

(A) $3$ ✓
(B) $\frac{1}{3}$
(C) $6$
(D) $5$

Worked solution

$m = \frac{\Delta y}{\Delta x}$ Rate of change is constant for a linear relationship

$m = \frac{11 - 5}{2 - 0} = \frac{6}{2}$ Use two rows, for example (0, 5) and (2, 11)

$m = 3$ y increases by 3 for each increase of 1 in x

Answer: $3$

For a linear relationship the rate of change is constant; divide the change in y by the corresponding change in x between any two rows, not just the change in y alone.

Question 10 · 2 pts

What is the slope of the line that passes through the points $(2, 5)$ and $(6, 13)$?

(A) $-\frac{1}{2}$
(B) $-2$
(C) $2$ ✓
(D) $\frac{1}{2}$

Worked solution

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$ Use the slope formula.

$m = \dfrac{13 - 5}{6 - 2}$ Substitute the two points.

$m = \dfrac{8}{4} = 2$ Simplify the fraction.

Answer: $2$

Slope is the change in $y$ divided by the change in $x$. The $y$-values increase by $8$ while the $x$-values increase by $4$, so the slope is $8 \div 4 = 2$. Reversing rise and run gives $\frac{1}{2}$, and subtracting in inconsistent order produces the negative distractors.

Question 1 · 2 pts

What is the slope of the line that passes through the points $(2, 3)$ and $(6, 11)$?

(A) $4$
(B) $-2$
(C) $\frac{1}{2}$
(D) $2$ ✓

Worked solution

$m = \frac{y_2 - y_1}{x_2 - x_1}$ Slope formula

$m = \frac{11 - 3}{6 - 2}$ Substitute the points

$m = \frac{8}{4} = 2$ Simplify

Answer: $2$

Slope is the change in $y$ divided by the change in $x$. Subtracting the coordinates in the same order gives $\frac{8}{4}$, which simplifies to $2$.

Question 4 · 2 pts

What is the slope of the line that passes through the points $(-2, 5)$ and $(4, -7)$?

(A) $-\dfrac{1}{2}$
(B) $-2$ ✓
(C) $2$
(D) $-6$

Worked solution

$m = \frac{y_2 - y_1}{x_2 - x_1}$ Slope formula.

$m = \frac{-7 - 5}{4 - (-2)}$ Substitute the two points.

$m = \frac{-12}{6}$

$m = -2$

Answer: $-2$

The slope is the change in $y$ divided by the change in $x$. Here the rise is $-12$ and the run is $6$, giving $-2$. Watch the double negative in $4 - (-2) = 6$ and keep the sign of the numerator.

Question 28 · 4 pts · constructed response

A hot air balloon begins descending at a constant rate. When the descent begins, the balloon is at a height of $240$ feet. Four minutes later, its height is $180$ feet. Determine the average rate of change of the balloon's height, in feet per minute, and explain what this value means in the context of the problem.

Model answer: The average rate of change is $-15$ feet per minute; the balloon's height decreases by 15 feet each minute.

Worked solution

$\text{rate}=\frac{180-240}{4-0}$ Use the change in height divided by the change in time.

$\text{rate}=\frac{-60}{4}=-15$ The rate is negative because the height is decreasing.

$\text{The balloon descends 15 feet every minute.}$ Interpret in context.

Answer: The average rate of change is $-15$ feet per minute; the balloon's height decreases by 15 feet each minute.

The average rate of change is the slope between the two points $(0, 240)$ and $(4, 180)$. Since the balloon loses 60 feet over 4 minutes, its height changes by $-15$ feet per minute, meaning it descends 15 feet each minute.

Question 26 · 4 pts · constructed response

At 2:00 p.m. the outdoor temperature was $68^\circ$F. At 7:00 p.m. the same day, the temperature was $53^\circ$F. Determine the average rate of change of the temperature, in degrees per hour, over this time interval. Explain what this value means in the context of the problem.

Model answer: $-3$ degrees per hour; on average, the temperature dropped $3^\circ$F each hour between 2:00 p.m. and 7:00 p.m.

Worked solution

$\text{change in temperature}=53-68=-15$ Final value minus initial value.

$\text{change in time}=7-2=5 \text{ hours}$

$\frac{-15}{5}=-3 \text{ degrees per hour}$ Average rate of change is change in temperature divided by change in time.

Answer: $-3$ degrees per hour; on average, the temperature dropped $3^\circ$F each hour between 2:00 p.m. and 7:00 p.m.

Average rate of change is the ratio of the change in the output (temperature) to the change in the input (time). Since the temperature fell 15 degrees over 5 hours, the rate is $-3$ degrees per hour, meaning the temperature decreased by about 3 degrees each hour.

Question 26 · 4 pts · constructed response

Determine the slope of the line that passes through the points $(-3, 7)$ and $(5, -9)$. Based on the slope, state whether the line is increasing or decreasing, and justify your answer.

Model answer: The slope is $-2$. The line is decreasing because the slope is negative, meaning $y$ decreases as $x$ increases.

Worked solution

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-9 - 7}{5 - (-3)}$ Apply the slope formula.

$m = \frac{-16}{8} = -2$

$\text{Since } m < 0, \text{ the line is decreasing.}$ A negative slope means y decreases as x increases.

Answer: The slope is $-2$. The line is decreasing because the slope is negative, meaning $y$ decreases as $x$ increases.

Slope measures the change in $y$ divided by the change in $x$ between two points. Here the change in $y$ is $-16$ over a change in $x$ of $8$, so $m = -2$. A negative slope always indicates a decreasing line.

Question 18 · 2 pts

On a long Saturday bike ride, Dave had traveled $0$ miles at hour $0$ and $55$ miles after $5.5$ hours. What was his average rate of change, in miles per hour, over the whole ride?

(A) $14.5$
(B) $11$
(C) $11.6$
(D) $10$ ✓

Worked solution

$\text{avg rate} = \frac{\Delta \text{distance}}{\Delta \text{time}}$ Average rate of change over the whole trip.

$= \frac{55 - 0}{5.5 - 0} = \frac{55}{5.5} = 10$ Endpoints only.

Answer: 10 \text{ mph}

Average rate of change uses only the start and end points: total change in y over total change in x. Individual segments can be faster or slower.

Question 29 · 2 pts · constructed response

Write an equation in slope-intercept form for the line passing through $(-2,5)$ with slope $-3$.

Model answer: y=-3x-1

Worked solution

$y=mx+b \Rightarrow 5=-3(-2)+b$ Substitute the point and slope to find b.

$5=6+b \Rightarrow b=-1$

$y=-3x-1$

Answer: y=-3x-1

Plug the point into y = mx + b to solve for the intercept b, then write the full equation.

Question 15 · 2 pts

What is the slope of the line whose equation is $4x + 2y = 10$?

(A) $-2$ ✓
(B) $-\frac{1}{2}$
(C) $4$
(D) $2$

Worked solution

$2y = -4x + 10$ Subtract 4x from both sides

$y = -2x + 5$ Divide every term by 2 to reach y = mx + b form

$m = -2$ The slope is the coefficient of x

Answer: $-2$

To find slope from a standard-form equation, solve for y so it is in $y = mx + b$ form; the slope is then the coefficient of x.

Question 16 · 2 pts

A plumber charges a flat service fee plus a constant hourly rate. A 1-hour job costs $90 and a 4-hour job costs $210. What is the plumber's hourly rate (the rate of change of cost with respect to time)?

(A) 120 dollars per hour
(B) 90 dollars per hour
(C) 40 dollars per hour ✓
(D) 30 dollars per hour

Worked solution

$\text{rate} = \frac{\Delta \text{cost}}{\Delta \text{time}}$ Rate of change equals change in output over change in input

$= \frac{210 - 90}{4 - 1} = \frac{120}{3}$ Use the points (1, 90) and (4, 210)

$= 40$ The cost rises 40 dollars per additional hour

Answer: 40 dollars per hour

The rate of change equals the change in output (cost) divided by the change in input (time); the flat service fee shifts the line but does not affect the slope.

Question 35 · 4 pts · constructed response

A candle burns at a constant rate. After $2$ hours of burning, the candle is $20$ centimeters tall. After $5$ hours of burning, the candle is $11$ centimeters tall. Determine the rate of change of the candle's height, including units. Then write a linear equation that models the height $h$, in centimeters, after $t$ hours, and state the candle's original height.

Model answer: The rate of change is $-3$ centimeters per hour. A model is $h = -3t + 26$, where $h$ is the height in centimeters after $t$ hours. The original height was $26$ centimeters.

Worked solution

$m = \dfrac{11 - 20}{5 - 2} = \dfrac{-9}{3} = -3$ Rate of change, in centimeters per hour.

$h - 20 = -3(t - 2)$ Point-slope form using the point $(2, 20)$.

$h = -3t + 6 + 20$ Distribute and simplify.

$h = -3t + 26$ Slope-intercept form.

$h(0) = -3(0) + 26 = 26$ Original height occurs when $t = 0$.

Answer: The rate of change is $-3$ centimeters per hour. A model is $h = -3t + 26$, where $h$ is the height in centimeters after $t$ hours. The original height was $26$ centimeters.

A constant burn rate means the situation is linear, so the rate of change is the slope found from the two given points. The negative slope shows the candle shrinks $3$ cm each hour, and evaluating the equation at $t = 0$ gives the starting height of $26$ cm. Both given data points satisfy $h = -3t + 26$.

Question 27 · 4 pts · constructed response

A line passes through the points $(2, 5)$ and $(6, 17)$. Determine the slope of the line, and write its equation in slope-intercept form.

Model answer: The slope is $3$, and the equation is $y = 3x - 1$.

Worked solution

$m = \dfrac{17 - 5}{6 - 2} = \dfrac{12}{4} = 3$ Apply the slope formula, change in y over change in x.

$5 = 3(2) + b$ Substitute the point $(2,5)$ and $m=3$ into $y = mx + b$.

$5 = 6 + b \Rightarrow b = -1$ Solve for the y-intercept.

$y = 3x - 1$ Check: $3(6) - 1 = 17$, which matches the second point.

Answer: The slope is $3$, and the equation is $y = 3x - 1$.

Slope is the ratio of the vertical change to the horizontal change between two points. Once the slope is known, substituting either given point into $y = mx + b$ lets you solve for the y-intercept and write the full equation.

Question 34 · 4 pts · constructed response

A candle burns at a constant rate. When it has been burning for 2 hours, its height is 21 centimeters. When it has been burning for 5 hours, its height is 12 centimeters. Write a linear equation that models the height $h$, in centimeters, of the candle after $t$ hours. State the rate of change, including units, and explain what it means in this situation.

Model answer: $h = -3t + 27$. The rate of change is $-3$ centimeters per hour, which means the candle's height decreases by 3 centimeters every hour.

Worked solution

$m = \dfrac{12 - 21}{5 - 2} = \dfrac{-9}{3} = -3$ Slope from the points (2, 21) and (5, 12).

$21 = -3(2) + b$ Substitute a known point to find the initial height b.

$21 = -6 + b \Rightarrow b = 27$ Solve for b.

$h = -3t + 27$ Linear model for the candle's height.

$\text{rate of change} = -3 \text{ cm per hour}$ The candle loses 3 cm of height each hour.

Answer: $h = -3t + 27$. The rate of change is $-3$ centimeters per hour, which means the candle's height decreases by 3 centimeters every hour.

The height changes at a constant rate, so the situation is linear. The slope, found from the two given points, is $-3$, and using a point gives the initial height $b = 27$. The negative rate of change tells us the candle burns down 3 centimeters for every hour it burns.

Question 35 · 4 pts · constructed response

Maria joined a gym that charges a one-time membership fee plus the same monthly charge every month. After 3 months, she had paid a total of $\$190$. After 7 months, she had paid a total of $\$310$. Determine the monthly charge, in dollars per month, and the one-time membership fee. Justify your answers.

Model answer: The monthly charge is $\$30$ per month and the one-time membership fee is $\$100$.

Worked solution

$m = \frac{310 - 190}{7 - 3} = \frac{120}{4} = 30$ The rate of change (slope) between the points $(3, 190)$ and $(7, 310)$ is the monthly charge.

$y = 30x + b$ The total cost is linear in the number of months $x$, with $b$ the membership fee.

$190 = 30(3) + b \;\Rightarrow\; b = 190 - 90 = 100$ Substitute the point $(3, 190)$.

$\text{Check: } 30(7) + 100 = 210 + 100 = 310$ The second point confirms the model.

Answer: The monthly charge is $\$30$ per month and the one-time membership fee is $\$100$.

Because the monthly charge is constant, the total paid is a linear function of the number of months. The slope between the two given points, $\frac{310 - 190}{7 - 3} = 30$, is the monthly charge, and substituting either point into $y = 30x + b$ gives the initial fee $b = 100$, the $y$-intercept of the model.

Question 4 · 2 pts

What is the slope of the line that passes through the points $(2, 5)$ and $(6, -3)$?

(A) $-2$ ✓
(B) $2$
(C) $\frac{1}{2}$
(D) $-\frac{1}{2}$

Worked solution

$m = \frac{y_2 - y_1}{x_2 - x_1}$ Use the slope formula.

$m = \frac{-3 - 5}{6 - 2} = \frac{-8}{4}$

$m = -2$

Answer: $-2$

Slope is the change in $y$ divided by the change in $x$. Here the $y$-values drop by 8 while the $x$-values increase by 4, so the slope is $-8/4 = -2$. Mixing up the order of subtraction in only one coordinate gives the sign-error distractor $2$, and inverting the ratio gives $\pm\frac{1}{2}$.

Question 35 · 4 pts · constructed response

A gym charges a one-time joining fee plus the same amount each month. After 3 months, a member has paid a total of $\$155$. After 7 months, the member has paid a total of $\$255$. Determine the rate of change of the total cost, in dollars per month, and explain what it means in the context of this situation. Then write an equation for the total cost $C$, in dollars, after $m$ months.

Model answer: The rate of change is $25$ dollars per month, meaning the gym charges $\$25$ each month. The equation is $C = 25m + 80$, where $\$80$ is the joining fee.

Worked solution

$\text{rate} = \frac{255 - 155}{7 - 3} = \frac{100}{4} = 25$ Slope is the change in total cost divided by the change in months.

$\text{Each month of membership costs } \$25.$ Interpret the slope in context.

$155 = 25(3) + b \Rightarrow b = 80$ Use the point (3, 155) to find the initial fee.

$C = 25m + 80$ This checks: C(7) = 25(7) + 80 = 255.

Answer: The rate of change is $25$ dollars per month, meaning the gym charges $\$25$ each month. The equation is $C = 25m + 80$, where $\$80$ is the joining fee.

The situation is linear because the monthly charge is constant, so the rate of change is the slope between the two given points, $\frac{100}{4} = 25$ dollars per month. Substituting one point into $C = 25m + b$ gives the y-intercept $80$, which represents the one-time joining fee.

Question 17 · 2 pts

For the exponential function $f(x) = 2^x$, what is the average rate of change over the interval $1 \le x \le 3$?

(A) $3$ ✓
(B) $2$
(C) $4$
(D) $6$

Worked solution

$f(1) = 2^1 = 2, \quad f(3) = 2^3 = 8$ Evaluate the function at the endpoints

$\text{avg rate} = \frac{f(3) - f(1)}{3 - 1}$ Average rate of change is the slope between the endpoints

$= \frac{8 - 2}{2} = \frac{6}{2} = 3$ Simplify

Answer: $3$

The average rate of change over an interval is the slope of the line joining the two endpoints: the change in $f(x)$ divided by the change in x.

Question 18 · 2 pts

Which of the following linear functions has the greatest rate of change?

(A) $y = -6x + 20$
(B) $y = 4x - 9$
(C) the line through $(1, 2)$ and $(3, 12)$ ✓
(D) the table with $x$: 0, 3, 6 and $y$: 1, 10, 19

Worked solution

$\text{(A) line through } (1,2),(3,12):\ m = \frac{12 - 2}{3 - 1} = \frac{10}{2} = 5$ Apply the slope formula

$\text{(B) } y = 4x - 9 \Rightarrow m = 4$ Slope-intercept form

$\text{(C) table: } m = \frac{10 - 1}{3 - 0} = \frac{9}{3} = 3$ Change in y over change in x

$\text{(D) } y = -6x + 20 \Rightarrow m = -6$ Slope-intercept form

$5 > 4 > 3 > -6$ The line through (1, 2) and (3, 12) has the greatest slope

Answer: the line through $(1, 2)$ and $(3, 12)$

Rate of change is the slope; the greatest rate of change is the largest slope value (most positive), not the largest number that appears or the steepest downward line.

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