Algebra and Linear Modeling Practice Problems

A set of vocabulary flashcards covering algebraic concepts, simple interest, linear modeling, intercepts, absolute value transformations, vertices, and quadratic function evaluation.

Simple Interest Formula

An interest model where interest is calculated solely on the original principal, represented by the formula $I = Prt$ or total accumulated value $A = P(1 + rt)$, where $P$ is the principal invested, $r$ is the annual simple interest rate (such as $3.5\%$), and $t$ is the investment duration in years (such as $4\,\text{years}$).

Standard Form of a Linear Equation

A linear equation structured as $Ax + By = C$, where $A$, $B$, and $C$ are integers, demonstrated by equations such as $6x + 3y = -12$.

Intercept Method for Graphing Lines

A graphing procedure where two points are found by determining the x-intercept (setting $y = 0$ to yield $(-2, 0)$) and the y-intercept (setting $x = 0$ to yield $(0, -4)$) for a linear equation like $6x + 3y = -12$, then connecting them on the coordinate plane.

Linear Function Modeling

A model representing a constant rate of change over time in the form $y = mt + b$, used to relate elapsed time to changing quantities, such as tracking pool water volume between $1{:}00\,\text{PM}$, $5{:}00\,\text{PM}$ ($10,400\,\text{gallons}$), and $9{:}00\,\text{PM}$.

Initial Value (y-intercept) in Linear Applications

The starting amount in a linear relationship when the input variable is zero ($t = 0$), corresponding to the quantity of water remaining in a pool at the moment refilling began at $1{:}00\,\text{PM}$.

Linear Earnings Function

A linear equation modeling total compensation with a fixed base value plus a per-unit rate, structured as $\text{Total Earnings} = \text{Base Salary} + (\text{Commission Rate}) \times c$, such as Tyler earning a base salary of $\$45,000$ per year plus $\$75$ per computer ($c$) sold toward an earnings goal of $\$51,000$.

Horizontal Shift of an Absolute Value Function

A transformation applied inside the absolute value bars $y = |x - h|$; replacing $x$ with $(x - 3)$ shifts the V-shaped graph $3\,\text{units}$ to the right, whereas replacing $x$ with $(x + 3)$ shifts it $3\,\text{units}$ to the left.

Vertical Shift of an Absolute Value Function

A transformation applied outside the absolute value bars $y = |x| + k$; adding a positive constant (such as $+3$) shifts the graph upward by $3\,\text{units}$, while subtracting a constant (such as $-3$) shifts the graph downward by $3\,\text{units}$.

Vertex of an Absolute Value Function

The central turning point $(h, k)$ of an absolute value graph written in the form $f(x) = a|x - h| + k$; for the function $f(x) = -2|x + 4| - 5$, the vertex is located at the coordinate pair $(-4, -5)$.

Function Evaluation

The algebraic procedure of replacing the input variable of a function with a specified number and computing the result; for example, evaluating $f(x) = 5x^2 - 9x + 18$ at $x = 3$ yields $f(3) = 5(3)^2 - 9(3) + 18 = 36$.