Linear Relationships Unit Review

Vocabulary practice flashcards covering slope, rate of change, initial value, similar triangles, and proportional relationships from the Linear Relationships Unit Review worksheet.

Rate of Change (Table Analysis)

The ratio of the vertical change to horizontal change in a table, calculated as $m = \frac{\Delta y}{\Delta x}$; for values $(-3, 10.5)$ and $(-2, 7)$, the rate of change is $\frac{7 - 10.5}{-2 - (-3)} = \frac{-3.5}{1} = -3.5$.

Slope of a Line from a Graph

The ratio of vertical rise to horizontal run between points on a grid, calculated as $m = \frac{\text{Rise}}{\text{Run}}$; for a line rising $5$ units over a run of $3$ units, the slope is $\frac{5}{3}$.

Rate of Change (Pottery Painting Studio)

The unit cost per person represented by the slope of a graphed line passing through $(2, 15)$ and $(4, 30)$, equal to $\frac{30 - 15}{4 - 2} = \frac{15}{2}$ (or $\$7.50$ per person).

Slope Between Points $(5, 4)$ and $(-4, 3)$

The constant rate found using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, evaluated as $m = \frac{3 - 4}{-4 - 5} = \frac{-1}{-9} = \frac{1}{9}$.

Slope Between Points $(10, 8)$ and $(9, 13)$

The constant rate found using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, evaluated as $m = \frac{13 - 8}{9 - 10} = \frac{5}{-1} = -5$.

Slope of a Vertical Line

A slope that is undefined because the horizontal run is zero ($\Delta x = 0$), as seen on a coordinate graph along the vertical line $x = 3$.

Point-Slope Verification

The process of testing candidate points against a given slope $m = 7$ and point $(3, 5)$; using the linear equation $y - 5 = 7(x - 3)$ or $y = 7x - 16$, the point $(-2, -30)$ lies on the line because $7(-2) - 16 = -30$.

Rate of Change from Points $(0, -10)$ and $(2, 20)$

The slope determined from a table of values containing initial coordinate $(0, -10)$ and coordinate $(2, 20)$, evaluated as $m = \frac{20 - (-10)}{2 - 0} = \frac{30}{2} = 15$.

Similar Triangles and Slope

The geometric principle establishing that triangles drawn under the same straight line are similar, meaning the ratio of vertical change to horizontal change is identical everywhere along the line, proving the slope of $\overline{AC}$ equals the slope of $\overline{DF}$.

Linear Relationship: Graph 10

A graph passing through $(0, 0)$ and $(5, 90)$ with slope $m = 18$, initial value $b = 0$, equation $y = 18x$, and classified as proportional.

Linear Relationship: Graph 11

A decreasing linear graph starting at $(0, 21)$ and passing through $(10, 15)$ with slope $m = -0.6$, initial value $b = 21$, equation $y = -0.6x + 21$, and classified as not proportional.

Linear Equation: $y = \frac{2}{7}x$

A linear relation with slope $m = \frac{2}{7}$ and initial value $b = 0$, which is proportional because it has a y-intercept of zero and passes through the origin.

Linear Equation: $y = -3.5x - 10$

A linear relation with slope $m = -3.5$ and initial value $b = -10$, which is not proportional because its y-intercept does not equal zero ($b \neq 0$).

Table Model: Problem 14

A dataset with pairs $(2, 20)$, $(4, 50)$, $(6, 80)$, and $(8, 110)$ having a slope of $m = 15$, an initial value of $b = -10$, an equation of $y = 15x - 10$, and classified as not proportional.

Kayla's Coffee Shop Earnings Equation

The linear equation $y = 6.25x + 8.50$, modeling an hourly rate of $\$6.25$ ($m = 6.25$) for $x$ hours worked plus a one-time tip amount of $\$8.50$ ($b = 8.50$) to calculate total earnings $y$.

Initial Value ($b$)

The starting value of the dependent variable when the independent variable is zero ($x = 0$), corresponding to the y-intercept of a line in the equation $y = mx + b$.

Proportional Relationship

A linear relationship where two quantities maintain a constant ratio, graphed as a straight line that passes directly through the origin $(0, 0)$ with an initial value of $b = 0$.