Pre-Calculus Chapter 2: Characteristics of Functions and Trigonometry

Vocabulary practice flashcards covering function characteristics (extrema, concavity, even/odd functions) and fundamental unit circle trigonometry (angles, trigonometric definitions, periodicity, and inverses) from Pre-Calculus Chapter 2.

Extrema

Points on a graph that are "high" points or "low" points, also referred to as extreme points.

Local Maximum

The largest value of a function within a small interval around an extreme point.

Local Minimum

The smallest value of a function within a small interval around an extreme point.

Global Maximum

The largest value of a function over its entire domain.

Global Minimum

The smallest value of a function over its entire domain.

Concave Up

A property of an interval of a graph where the curve forms an upright cup or U-shape, meaning the slopes of consecutive segments are increasing throughout the interval.

Concave Down

A property of an interval of a graph where the slopes of consecutive segments are decreasing throughout the interval.

Point of Inflection

A point on a graph where the concavity changes from concave up to concave down or from concave down to concave up.

Even Function

A function that satisfies $f(-x) = f(x)$ for all values in its domain, displaying symmetry across the y-axis.

Odd Function

A function that satisfies $f(-x) = -f(x)$ for all values in its domain, displaying rotational symmetry about the origin.

Reference Angle

The positive acute angle formed between the terminal ray of an angle and the x-axis.

Coterminal Angles

Two or more angles in standard position that share the exact same initial side and terminal side.

Unit Circle

A circle centered at the origin $(0, 0)$ with a radius of $1$, modeled by the equation $x^2 + y^2 = 1$.

Sine ($\sin(\theta)$)

For any angle $\theta$, the y-coordinate of the point on the unit circle reached by a rotation of $\theta$ radians counterclockwise from the positive x-axis.

Cosine ($\cos(\theta)$)

For any angle $\theta$, the x-coordinate of the point on the unit circle reached by a rotation of $\theta$ radians counterclockwise from the positive x-axis.

Tangent ($\tan(\theta)$)

The slope of the terminal ray of an angle formed by a rotation of $\theta$ radians from standard position, given by $\frac{y}{x}$ or $\frac{\sin(\theta)}{\cos(\theta)}$.

Pythagorean Identity

The trigonometric relationship $\sin^2(\theta) + \cos^2(\theta) = 1$, derived from the legs and unit hypotenuse of a right triangle within the unit circle.

Periodic Function

A function whose graph repeats itself identically over regular intervals as it is followed from left to right.

Period

The smallest horizontal shift possible that preserves a periodic function's appearance, representing the length of one full cycle.

Amplitude

Half the distance between the maximum and minimum values of a periodic sinusoidal wave.

Inverse Sine ($\arcsin(x)$)

The inverse relation of $\sin(x)$ with its domain restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$, resulting in a domain of $[-1, 1]$ and a range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Inverse Cosine ($\arccos(x)$)

The inverse relation of $\cos(x)$ with its domain restricted to $[0, \pi]$, resulting in a domain of $[-1, 1]$ and a range of $[0, \pi]$.