Algebra 2 Flashcards: Recognize Constant Rate Changes

What this deck covers

This deck focuses on Recognize Constant Rate Changes, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the constant speed if distance is $60$ miles in $2$ hours at steady speed.

Answer: $30$ miles per hour. Rate equals distance divided by time: $\frac{60}{2}=30$.

Flashcard 2: What table feature confirms a constant rate when $x$ increases by $1$ each row?

Answer: The first differences of $y$ are constant. Equal spacing in $x$ with constant $y$ differences confirms linearity.

Flashcard 3: What phrase most strongly signals constant rate: "per" or "percent"?

Answer: "Per" (constant additive rate per unit). "Per" indicates additive rate; "percent" indicates multiplicative rate.

Flashcard 4: Identify whether the data show constant rate: $x:0,1,2$ and $y:5,9,16$.

Answer: No, first differences are not constant. Differences are $4$ then $7$, which are not equal.

Flashcard 5: What does constant rate imply about $\Delta y$ when $\Delta x$ doubles?

Answer: $\Delta y$ doubles as well. Constant rate means proportional relationship between $\Delta x$ and $\Delta y$.

Flashcard 6: Identify the constant rate of change for $y=7x-3$.

Answer: $7$. The coefficient of $x$ is the rate of change.

Flashcard 7: Identify whether the situation is constant rate: $y$ doubles whenever $x$ increases by $1$.

Answer: No, that is exponential (not constant difference). Doubling is multiplicative change, not constant additive change.

Flashcard 8: What does a negative constant rate of change mean about $y$ as $x$ increases?

Answer: $y$ decreases by a constant amount per unit increase in $x$. Negative slope means $y$ decreases as $x$ increases.

Flashcard 9: Which description matches constant rate: "$y$ changes by a fixed amount" or "$y$ changes by a fixed factor"?

Answer: "$y$ changes by a fixed amount". Fixed amount indicates additive; fixed factor indicates multiplicative.

Flashcard 10: What must be true about slopes between any two pairs of points on a line?

Answer: They are equal (constant slope). All points on a line have the same slope between any pair.

Flashcard 11: What is the constant rate of change in $y=2+4x$?

Answer: $4$. The coefficient of $x$ gives the rate of change.

Flashcard 12: What is the name of the constant rate of change in a linear function $y=mx+b$?

Answer: The slope $m$. The coefficient of $x$ represents the constant rate.

Flashcard 13: What is the constant rate of change for the proportional relationship $y=5x$?

Answer: $5$. In $y=kx$, the constant $k$ is the rate.

Flashcard 14: Identify the constant rate in the verbal rule: "Start at $10$ and add $2$ each day."

Answer: $2$ per day. The additive change per time unit.

Flashcard 15: What is the constant rate of change for the line $3x-2y=8$?

Answer: $\frac{3}{2}$. Rearranging to $y=\frac{3}{2}x-4$ shows slope $\frac{3}{2}$.

Flashcard 16: Find the rate of change from $(2,5)$ to $(6,13)$.

Answer: $m=\frac{13-5}{6-2}=2$. Using slope formula: $\frac{13-5}{6-2}=\frac{8}{4}=2$.

Flashcard 17: What does $m=0$ mean about the rate of change in $y=mx+b$?

Answer: No change; $y$ is constant. Zero slope means $y$ stays the same regardless of $x$.

Flashcard 18: Identify whether $y=x^2+1$ has a constant rate of change.

Answer: No, the rate of change is not constant. Quadratic functions have changing rates of change.

Flashcard 19: Identify whether the situation is constant rate: $y$ increases by $3$ whenever $x$ increases by $1$.

Answer: Yes, constant rate ($m=3$). Fixed increase of $3$ for each unit of $x$ means constant rate.

Flashcard 20: What does "constant percent rate" usually indicate: linear or exponential change?

Answer: Exponential change. Constant percent growth creates exponential patterns.

Flashcard 21: Identify whether the sequence rule $y=2+4x$ represents constant change per step in $x$.

Answer: Yes, constant change of $4$ per $1$ in $x$. Linear form with coefficient $4$ as the constant rate.

Flashcard 22: What graph shape indicates a constant rate of change between $x$ and $y$?

Answer: A straight line (linear relationship). Linear functions have constant slope, creating straight lines.

Flashcard 23: Identify the initial value in "Start at $10$ and add $2$ each day" for $y$ at day $0$.

Answer: $10$. The starting value when the independent variable equals zero.

Flashcard 24: Choose the model type for constant rate: $y=mx+b$ or $y=a\cdot b^x$.

Answer: $y=mx+b$. Linear models have constant rates; exponential models have constant ratios.

Flashcard 25: Find the slope of the line through $(1,9)$ and $(5,1)$.

Answer: $m=\frac{1-9}{5-1}=-2$. Using slope formula: $\frac{1-9}{5-1}=\frac{-8}{4}=-2$.

Flashcard 26: What does the phrase "per unit" refer to in a constant rate of change?

Answer: The change in one variable for $\Delta x=1$. Rate of change when $x$ increases by exactly $1$.

Flashcard 27: What is the constant rate if $y$ always increases by $10$ when $x$ increases by $2$?

Answer: $5$ per unit $x$. Rate per unit: $\frac{10}{2}=5$ per unit $x$.

Flashcard 28: What is the constant rate if $x:0,1,2$ and $y:5,8,11$?

Answer: $3$ per unit $x$. Change in $y$ divided by change in $x$: $\frac{3}{1}=3$.

Flashcard 29: Identify whether this indicates constant rate: When $x$ increases by $2$, $y$ always increases by $10$.

Answer: Yes, constant rate ($m=5$). Fixed change of $10$ for every $2$ in $x$ means constant rate.

Flashcard 30: Identify the constant rate if $y$ increases by $15$ when $x$ increases by $5$ (same every time).

Answer: $3$ per unit $x$. Rate equals total change divided by input change: $\frac{15}{5}=3$.

Flashcard 31: Which situation represents constant rate: "$60$ miles in $2$ hours at steady speed" or "speeding up"?

Answer: "$60$ miles in $2$ hours at steady speed". Steady speed means constant rate; speeding up means changing rate.

Flashcard 32: What formula gives the constant rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using two points.

Flashcard 33: Identify whether the points $(0,2)$, $(2,6)$, $(4,10)$ show a constant rate of change.

Answer: Yes, constant rate ($m=2$). All consecutive slopes equal $2$, confirming constant rate.

Flashcard 34: Which is constant rate: $y$ changes by $-6$ when $x$ changes by $3$, or by $-7$ when $x$ changes by $3$?

Answer: The $-6$ for $\Delta x=3$ case (rate $-2$ per $1$). Rate of $-2$ per unit gives $-6$ change for $\Delta x=3$.

Flashcard 35: Which change pattern indicates constant rate: constant differences in $y$ or constant ratios in $y$?

Answer: Constant differences in $y$ (for equal $\Delta x$). Constant differences indicate linear; constant ratios indicate exponential.

Flashcard 36: Which slope indicates constant decrease of $5$ per $1$ in $x$: $m=-5$ or $m=5$?

Answer: $m=-5$. Negative slope means decrease of $5$ per unit increase in $x$.

Flashcard 37: Identify whether $y=4x+1$ has a constant rate of change.

Answer: Yes, constant rate ($m=4$). Linear functions have constant slope everywhere.

Flashcard 38: Identify whether $3x-2y=8$ represents a constant rate of change relationship.

Answer: Yes, it is linear (constant slope). Linear equations represent constant rate relationships.

Flashcard 39: Find the rate of change from $(-1,4)$ to $(3,-8)$.

Answer: $m=\frac{-8-4}{3-(-1)}=-3$. Using slope formula: $\frac{-8-4}{3-(-1)}=\frac{-12}{4}=-3$.

Flashcard 40: Identify whether the data show constant rate: $x:0,1,2$ and $y:5,8,11$.

Answer: Yes, constant rate ($\Delta y=3$ each step). Equal differences of $3$ between consecutive $y$-values.

Flashcard 41: Identify whether the points $(0,1)$, $(1,2)$, $(2,4)$ show a constant rate of change.

Answer: No, slopes differ ($1$ then $2$). Slopes change from $1$ to $2$, so rate is not constant.

Flashcard 42: What does a constant ratio $\frac{y}{x}$ (with $x\neq 0$) indicate about $y$ versus $x$?

Answer: A proportional linear relationship $y=kx$. Constant ratio means $y$ is proportional to $x$.

Flashcard 43: Which equation form explicitly shows constant rate and initial value: $y=mx+b$ or $y=ax^2+c$?

Answer: $y=mx+b$. Linear form shows constant rate $m$ and starting value $b$.

Flashcard 44: Compute the unit rate if $\Delta y=-6$ when $\Delta x=3$.

Answer: $\frac{\Delta y}{\Delta x}=-2$. Divide change in $y$ by change in $x$: $\frac{-6}{3}=-2$.

Flashcard 45: Identify the constant rate of change for $y=-\frac{3}{2}x+8$.

Answer: $-\frac{3}{2}$. The coefficient of $x$ is the rate of change.

Flashcard 46: Which statement indicates constant rate: "adds $4$ each hour" or "multiplies by $4$ each hour"?

Answer: "Adds $4$ each hour". Addition indicates constant rate, multiplication indicates exponential.

Flashcard 47: Identify whether this is constant rate: A population increases by $5\%$ each year.

Answer: No, that is exponential (constant percent change). Percent change indicates exponential, not linear growth.

Flashcard 48: What does it mean for $y$ to change at a constant rate per unit of $x$?

Answer: $\frac{\Delta y}{\Delta x}$ is constant for equal $\Delta x$ intervals. The ratio of change in $y$ to change in $x$ stays the same.

Flashcard 49: Identify whether this is constant rate: A tank loses $3$ liters every minute.

Answer: Yes, constant rate ($-3$ liters per minute). Fixed amount lost per time unit indicates constant rate.

Flashcard 50: Which relationship has constant rate: $y=x+3$ or $y=3^x$?

Answer: $y=x+3$. Linear has constant rate; exponential has constant ratio.