Linear Functions and Slope-Intercept Form

Comprehensive vocabulary flashcards covering linear relationships, deriving slope-intercept form ($$y = mx + b$$), analyzing tables and graphs, and evaluating real-world cost and revenue models.

Slope-Intercept Form

The linear equation written as $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

Slope Formula

The algebraic ratio $m = \frac{y_2 - y_1}{x_2 - x_1}$, used to calculate the rate of change or steepness between two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$.

Y-Intercept

The coordinate point $(0, b)$ where a graph crosses the vertical y-axis, representing the initial value or constant term in $y = mx + b$.

Base Fee (Linear Context)

A fixed initial charge—such as the starting $\$5$ fee in a bike rental scenario—that corresponds to the y-intercept where the independent variable (time) equals zero.

Initial Amount (Revenue Modeling)

The fixed upfront earnings Mateo receives to set up his food truck before selling any individual items, modeled mathematically by the y-intercept of the revenue line.

Slope Triangles

Right triangles constructed along a line whose corresponding side lengths have constant ratios, showing that the slope remains identical across every segment of the line.

Denominator Cancellation Strategy

The method of selecting an $x$-value equal to the denominator of a fractional slope (such as choosing $x = 4$ for $y = \frac{7}{4}x + 1$) so that numerator and denominator cancel out, simplifying the calculation to an integer result.

Graphing $y = 3x - 2$

Plotting a straight line with a y-intercept of $(0, -2)$ and a slope of $3$ (or $\frac{3}{1}$), rising $3$ units vertically for every $1$ unit moved horizontally to the right.

Line with Slope $-2$ and Y-Intercept $-4$

A line represented by the equation $y = -2x - 4$, which crosses the y-axis at $(0, -4)$ and decreases by $2$ units vertically for every $1$ unit moved to the right.

Linear Equation from a Table (DeltaMath #20B)

The slope-intercept equation $y = 8x - 5$, determined by finding the y-intercept $b = -5$ when $x = 0$ and computing the constant rate of change $m = \frac{3 - (-5)}{1 - 0} = 8$.

Linear Equation from a Graph (DeltaMath #20B)

The fully simplified slope-intercept equation $y = -2x + 4$, identified from a graphed line that intersects the y-axis at $(0, 4)$ and descends $2$ units for every $1$ unit of horizontal run.

Fully Simplified Form

An algebraic expression or equation in which all fractions are reduced to lowest terms, like terms are combined, and signs are simplified without redundant parentheses or operators.