Trigonometry, Geometric Transformations, Inequalities, and Basic Statistics Vocabulary

Vocabulary flashcards generated from lecture notes covering trigonometry laws, geometric transformations, quadratic/absolute value inequalities, and descriptive statistics.

Law of Sines

A trigonometric rule used when an angle and its opposite side are known, expressed as $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$, which applies to AAS, ASA, and SSA cases.

Ambiguous Case (SSA)

A specific condition when applying the Law of Sines that can result in 0, 1, or 2 possible valid triangles.

Law of Cosines

A trigonometric rule used when an angle is trapped between two sides or when all three sides are given, expressed as $a^2 = b^2 + c^2 - 2bc \cos(A)$, $b^2 = a^2 + c^2 - 2ac \cos(B)$, and $c^2 = a^2 + b^2 - 2ab \cos(C)$, which applies to SAS and SSS cases.

Triangle Sum Rule

The geometric principle used for solving interior angles, stating that $A + B + C = 180^\circ$.

Translation

A geometric transformation that involves sliding a point $(x, y)$ to $(x+h, y+k)$.

Reflection across x-axis

A geometric transformation that maps a point $(x, y)$ to $(x, -y)$.

Reflection across y-axis

A geometric transformation that maps a point $(x, y)$ to $(-x, y)$.

Dilation

A geometric transformation that scales a shape's size up or down without changing its shape.

Critical Points

Roots found by factoring $ax^2 + bx + c = 0$ to establish boundaries for quadratic inequalities.

Boundary Curves

Graphical lines for inequalities, where strict inequalities ($<, >$) are drawn as dashed lines/curves and non-strict inequalities ($\le, \ge$) are drawn as solid lines/curves.

Test Point Method

A procedure where a point such as $(0,0)$ is substituted into an inequality to test if a region is included in the solution set.

Absolute Value

The measurement of distance from zero on a number line, denoted as $|a|$, which is always non-negative.

Absolute Value Negative Rule

The rule stating that an absolute value expression set equal to or less than a negative number has no solution.

Quartiles (Q)

Positional values that divide an ordered data set into 4 equal parts, representing 25% increments per quartile.

Deciles (D)

Positional values that divide an ordered data set into 10 equal parts, representing 10% increments per decile.

Percentiles (P)

Positional values that divide an ordered data set into 100 equal parts, representing 1% increments per percentile.

Positional Equivalencies

The relationship where $Q_2 = D_5 = P_{50} = \text{Median}$.

Outliers

Data values that fall significantly above or below the general trend of the dataset.

Interquartile Range (IQR)

A statistical measure of dispersion calculated as $\text{IQR} = Q_3 - Q_1$.

Lower Fence

The cutoff boundary for lower outliers, calculated using the formula $Q_1 - 1.5(\text{IQR})$.

Upper Fence

The cutoff boundary for upper outliers, calculated using the formula $Q_3 + 1.5(\text{IQR})$.

Box-and-Whisker Plot

A graphical display of the 5-number summary consisting of Minimum, $Q_1$, Median, $Q_3$, and Maximum.

Frequency Polygon

A line graph created by plotting interval midpoints against their corresponding frequencies.

Cumulative Frequency Polygon (Ogive)

A graph created by plotting upper class boundaries against cumulative frequencies.

Class Midpoint (Class Mark)

The center value of a class interval, calculated as $\frac{\text{Lower Limit} + \text{Upper Limit}}{2}$.

Exact Class Boundaries

Interval boundaries adjusted for continuous data by subtracting $0.5$ from the lower limit and adding $0.5$ to the upper limit.