Algebra 2 Flashcards: Interpreting Parameters In Linear Exponential Models

What this deck covers

This deck focuses on Interpreting Parameters In Linear Exponential Models, 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: What does the $y$-intercept represent on a graph of a contextual linear function?

Answer: It represents the starting amount at $x = 0$. This is where the line crosses the y-axis at $x = 0$.

Flashcard 2: What condition on $b$ indicates exponential decay in $y = a \, b^x$?

Answer: Decay occurs when $0 < b < 1$. Values between 0 and 1 multiply to decrease the quantity.

Flashcard 3: What does the parameter $b$ represent in an exponential model $y = a \, b^x$ in context?

Answer: $b$ is the growth factor per $1$ unit increase in $x$. Each unit increase in $x$ multiplies the output by this factor.

Flashcard 4: What does the parameter $b$ represent in a linear model $y = mx + b$ in context?

Answer: $b$ is the initial value, the output when $x = 0$. The y-intercept shows the starting value before any input changes.

Flashcard 5: What does the parameter $m$ represent in a linear model $y = mx + b$ in context?

Answer: $m$ is the rate of change (slope), in $y$-units per $x$-unit. The slope quantifies how much the output changes per unit input change.

Flashcard 6: Identify the meaning of $a = 2000$ in $P(t) = 2000(1.05)^t$ for a population model.

Answer: Initial population is $2000$ at $t = 0$. The coefficient represents the starting population count.

Flashcard 7: Identify the meaning of $0.6$ in $V(t) = 80(0.6)^t$ for a value-depreciation model.

Answer: Value keeps $60\%$ each time unit (a $40\%$ decrease each unit). The base shows the remaining fraction after each depreciation period.

Flashcard 8: Identify the growth/decay factor for $y = 120(0.85)^t$ in context.

Answer: Factor is $0.85$, meaning multiply by $0.85$ each time unit. The base shows how the quantity changes each time period.

Flashcard 9: What is the percent change per unit for $y = 500(1.03)^t$?

Answer: $3\%$ growth per time unit. Convert to percent: $(1.03 - 1) \times 100\% = 3\%$ growth.

Flashcard 10: What is the percent change per unit for $y = 500(0.97)^t$?

Answer: $3\%$ decay per time unit. Convert to percent: $(1 - 0.97) \times 100\% = 3\%$ decay.

Flashcard 11: Identify the meaning of $b = 120$ in $y = 65t + 120$ if $t$ is hours and $y$ is miles.

Answer: The starting distance is $120$ miles at $t = 0$. The y-intercept represents the initial distance position.

Flashcard 12: What does the parameter $k$ represent in $y = a \, b^{x-k}$ in context?

Answer: $k$ shifts the input: the reference point occurs at $x = k$. The horizontal shift changes when the reference value occurs.

Flashcard 13: What does the parameter $(1+r)$ represent in $y = a(1+r)^t$?

Answer: $(1+r)$ is the multiplier applied each time unit. This factor determines how the quantity changes per time unit.

Flashcard 14: What does the parameter $a$ represent in $y = a(1+r)^t$ when $t$ is time?

Answer: $a$ is the amount at time $t = 0$. This represents the initial quantity before any time passes.

Flashcard 15: What does the parameter $c$ represent in $y = mx + b + c$ in context?

Answer: $c$ is a vertical shift, adding $c$ to every output value. Vertical shifts move all output values up or down by $c$.

Flashcard 16: What does the parameter $h$ represent in $y = m(x-h) + b$ in context?

Answer: $h$ shifts the input: the value $b$ occurs at $x = h$. The horizontal shift changes when the y-intercept value occurs.

Flashcard 17: What does the difference $y(x+1) - y(x)$ equal for $y = mx + b$?

Answer: The difference is constant and equals $m$. Consecutive outputs have a constant additive relationship.

Flashcard 18: What does the $x$-intercept represent on a graph of a contextual linear function?

Answer: It is when the output equals $0$ (the quantity reaches $0$). This is where the line crosses the x-axis at $y = 0$.

Flashcard 19: What does the $y$-intercept represent on a graph of a contextual linear function?

Answer: It represents the starting amount at $x = 0$. This is where the line crosses the y-axis at $x = 0$.

Flashcard 20: What does a positive slope $m>0$ mean in a contextual linear model $y = mx + b$?

Answer: The quantity increases by $m$ per $1$ unit increase in $x$. Positive slopes indicate an increase in the dependent variable.

Flashcard 21: What does a negative slope $m<0$ mean in a contextual linear model $y = mx + b$?

Answer: The quantity decreases by $|m|$ per $1$ unit increase in $x$. Negative slopes indicate a decrease in the dependent variable.

Flashcard 22: What condition on $b$ indicates exponential growth in $y = a \, b^x$?

Answer: Growth occurs when $b > 1$. Values greater than 1 multiply to increase the quantity.

Flashcard 23: What does $r$ represent in $y = a(1-r)^x$ when $0<r<1$?

Answer: $r$ is the percent decay rate per $x$-unit, written as a decimal. Decay rate as a decimal: $0.93$ means $7\%$ decay.

Flashcard 24: What does $r$ represent in $y = a(1+r)^x$ when $r$ is given as a decimal?

Answer: $r$ is the percent rate per $x$-unit, written as a decimal. Growth rate as a decimal: $1.08$ means $8\%$ growth.

Flashcard 25: What does the parameter $a$ represent in an exponential model $y = a \, b^x$ in context?

Answer: $a$ is the initial value, the output when $x = 0$. This is the y-intercept, representing the starting amount.

Flashcard 26: Identify the meaning of $m = 65$ in $y = 65t + 120$ if $t$ is hours and $y$ is miles.

Answer: The speed is $65$ miles per hour. The slope represents the rate of change in distance per time.

Flashcard 27: Identify the growth/decay factor for $y = 120(0.85)^t$ in context.

Answer: Factor is $0.85$, meaning multiply by $0.85$ each time unit. The base shows how the quantity changes each time period.

Flashcard 28: Identify the initial value for $y = 120(0.85)^t$ in context.

Answer: Initial value is $120$ at $t = 0$. The coefficient of the exponential term at $t = 0$.

Flashcard 29: Identify the rate of change for $y = -2x + 9$ in context.

Answer: Rate of change is $-2$ per $1$ unit of $x$. The coefficient of $x$ shows the change per unit input.

Flashcard 30: Identify the initial value for $y = 3x + 50$ in context.

Answer: Initial value is $50$ (the value when $x = 0$). The constant term is the value when the input is zero.