| Term | Definition |
|---|---|
| concavity | The curvature of a function, describing whether the graph curves upward (concave up) or downward (concave down). |
| cycle | A single complete repetition of a periodic pattern that can be used to construct the entire graph of a periodic relationship. |
| intervals of decrease | Sections of a function's domain where the output values are getting smaller as the input increases. |
| intervals of increase | Sections of a function's domain where the output values are getting larger as the input increases. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| periodic function | A function that repeats its values at regular intervals, where f(x+k) = f(x) for all x in the domain, with k being the period. |
| periodic relationship | A relationship between two variables where output values demonstrate a repeating pattern over successive equal-length intervals as input values increase. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| repeating pattern | A sequence of values or behaviors that recurs at regular intervals in a periodic relationship. |
| Term | Definition |
|---|---|
| domain restrictions | Limitations on the input values of a function based on mathematical validity, contextual meaning, or extreme values in the data set. |
| inverse trigonometric functions | Functions that reverse the operation of trigonometric functions, taking a trigonometric ratio as input and returning an angle measure as output. |
| periodic | A property of trigonometric functions where they repeat their values at regular intervals. |
| trigonometric equations | Equations that contain trigonometric functions and require finding the values of the variable that satisfy the equation. |
| trigonometric inequalities | Inequalities that contain trigonometric functions and require finding the values of the variable that satisfy the inequality. |
| Term | Definition |
|---|---|
| cosecant function | The reciprocal of the sine function, denoted f(θ) = csc θ, defined where sin θ ≠ 0. |
| cotangent function | The reciprocal of the tangent function, denoted f(θ) = cot θ, equivalent to cos θ/sin θ where sin θ ≠ 0. |
| range | The set of all possible output values that a function can produce. |
| reciprocal function | A function formed by taking the reciprocal (1/f) of another function. |
| secant function | The reciprocal of the cosine function, denoted f(θ) = sec θ, defined where cos θ ≠ 0. |
| vertical asymptotes | Lines where a function approaches infinity; for secant and cosecant functions, these occur where cosine and sine equal zero, respectively. |
| Term | Definition |
|---|---|
| algebraic manipulation | The process of rewriting expressions using algebraic operations to transform them into equivalent forms. |
| cosine sum identity | The trigonometric identity cos(α + β) = cos α cos β - sin α sin β, which expresses the cosine of a sum of two angles. |
| difference identities | Trigonometric identities derived from sum identities by substituting negative angles, used to express trigonometric functions of angle differences. |
| domain restrictions | Limitations on the input values of a function based on mathematical validity, contextual meaning, or extreme values in the data set. |
| double-angle identities | Trigonometric identities that express trigonometric functions of twice an angle in terms of functions of the original angle. |
| equivalent analytic representations | Different algebraic forms of trigonometric expressions that are mathematically equal and can reveal different properties or simplify problem-solving. |
| equivalent forms | Different ways of writing the same mathematical expression that have equal values for all valid inputs. |
| Pythagorean identity | The fundamental trigonometric identity sin² θ + cos² θ = 1, derived from the Pythagorean Theorem applied to the unit circle. |
| sine sum identity | The trigonometric identity sin(α + β) = sin α cos β + cos α sin β, which expresses the sine of a sum of two angles. |
| trigonometric equations | Equations that contain trigonometric functions and require finding the values of the variable that satisfy the equation. |
| trigonometric expressions | Mathematical expressions involving trigonometric functions such as sine, cosine, tangent, and their reciprocals. |
| trigonometric identity | An equation involving trigonometric functions that is true for all values in the domain of the functions. |
| trigonometric inequalities | Inequalities that contain trigonometric functions and require finding the values of the variable that satisfy the inequality. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| Term | Definition |
|---|---|
| angle in standard position | An angle positioned in the coordinate plane with its vertex at the origin and one ray coinciding with the positive x-axis. |
| complex number | A number of the form a + bi, where a and b are real numbers and i is the imaginary unit. |
| complex plane | A coordinate system where complex numbers are represented as points, with the real part on the horizontal axis and the imaginary part on the vertical axis. |
| origin | The central point in a polar coordinate system from which all distances (radii) are measured. |
| polar coordinate | A coordinate system in which points are located by their distance from the origin (radius r) and their angle measure (θ) from the positive x-axis. |
| polar coordinate system | A coordinate system based on circles centered at the origin and lines through the origin, where points are located using an ordered pair (r, θ). |
| radius | In polar coordinates, the distance from the origin to a point, represented by |r|. |
| rectangular coordinate | An ordered pair (x, y) representing the horizontal and vertical position of a point in the plane. |
| rectangular coordinate system | A coordinate system where points are located using an ordered pair (x, y) representing horizontal and vertical distances from the origin. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| Term | Definition |
|---|---|
| angle measure | The input value in a polar function that represents the direction from the positive x-axis, typically measured in radians or degrees. |
| domain | The set of all possible input values for which a function is defined. |
| origin | The central point in a polar coordinate system from which all distances (radii) are measured. |
| polar coordinate | A coordinate system in which points are located by their distance from the origin (radius r) and their angle measure (θ) from the positive x-axis. |
| polar functions | Functions of the form r = f(θ) where the input is an angle measure and the output is a radius, used to create graphs in polar coordinates. |
| positive x-axis | The reference direction in a polar coordinate system from which angle measures are taken. |
| radius | In polar coordinates, the distance from the origin to a point, represented by |r|. |
| Term | Definition |
|---|---|
| average rate of change | The change in the output of a function divided by the change in the input over a specified interval, calculated as (f(b) - f(a))/(b - a) for the interval [a, b]. |
| polar function | A function of the form r = f(θ) that describes a curve in the polar coordinate system, where r is the distance from the origin and θ is the angle. |
| relative extremum | A point on a function where the function changes from increasing to decreasing (relative maximum) or from decreasing to increasing (relative minimum). |
| Term | Definition |
|---|---|
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| radian measure | The measure of an angle defined as the ratio of the arc length subtended by the angle to the radius of the circle. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| standard position | The position of an angle with its vertex at the origin and its initial side along the positive x-axis. |
| tangent function | A trigonometric function, denoted f(θ) = tan θ, that gives the slope of the terminal ray of an angle in standard position on the unit circle. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| Term | Definition |
|---|---|
| angle in standard position | An angle positioned in the coordinate plane with its vertex at the origin and one ray coinciding with the positive x-axis. |
| circle centered at the origin | A circle whose center is at the point (0, 0) on the coordinate plane. |
| coordinates | The ordered pair (x, y) that specifies the location of a point on a coordinate plane. |
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| equilateral triangle | A triangle with all three sides equal in length and all angles measuring 60°. |
| isosceles right triangle | A right triangle with two equal sides and angles of 45°-45°-90°. |
| multiples of π/4 | Angles that are integer multiples of π/4 radians (45°), including π/4, π/2, 3π/4, π, etc. |
| multiples of π/6 | Angles that are integer multiples of π/6 radians (30°), including π/6, π/3, π/2, 2π/3, etc. |
| quadrant | One of the four regions of the coordinate plane divided by the x-axis and y-axis. |
| radian | A unit of angle measure where one radian is the angle formed when the arc length equals the radius of the circle. |
| radius | In polar coordinates, the distance from the origin to a point, represented by |r|. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| Term | Definition |
|---|---|
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| domain | The set of all possible input values for which a function is defined. |
| oscillate | To move back and forth between two values in a regular, repeating pattern. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| standard position | The position of an angle with its vertex at the origin and its initial side along the positive x-axis. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| x-coordinate | The horizontal position of a point, representing its distance from the y-axis. |
| y-coordinate | The vertical position of a point, representing its distance from the x-axis. |
| Term | Definition |
|---|---|
| amplitude | The absolute value of the coefficient a in a sinusoidal function, representing the maximum distance from the midline to the peak or trough of the graph. |
| concave down | A characteristic of a graph where the rate of change is decreasing, creating a curve that opens downward. |
| concave up | A characteristic of a graph where the rate of change is increasing, creating a curve that opens upward. |
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| even function | A function that is graphically symmetric over the line x = 0 and satisfies the property f(−x) = f(x). |
| frequency | The number of complete cycles of a sinusoidal function that occur over a unit interval of input values. |
| midline | The horizontal line around which a sinusoidal function oscillates, located at y = d in the function a sin(b(θ + c)) + d. |
| odd function | A function that is graphically symmetric about the point (0,0) and satisfies the property f(−x) = −f(x). |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| reflective symmetry | A property of a graph that is mirror-symmetric across a line; the cosine function has reflective symmetry over the y-axis. |
| rotational symmetry | A property of a graph that looks the same when rotated 180 degrees about a point; the sine function has rotational symmetry about the origin. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| sinusoidal function | A function of the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. |
| Term | Definition |
|---|---|
| amplitude | The absolute value of the coefficient a in a sinusoidal function, representing the maximum distance from the midline to the peak or trough of the graph. |
| horizontal dilation | A transformation that stretches or compresses the graph of a function horizontally by multiplying the input by a constant factor b, written as g(x) = f(bx). |
| horizontal translation | A transformation that shifts the graph of a function left or right by adding a constant to the input, written as g(x) = f(x + h). |
| midline | The horizontal line around which a sinusoidal function oscillates, located at y = d in the function a sin(b(θ + c)) + d. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| phase shift | A horizontal translation of a sinusoidal function represented by the constant c, which shifts the graph left or right by -c units. |
| sinusoidal function | A function of the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. |
| vertical dilation | A transformation that stretches or compresses the graph of a function vertically by multiplying the function by a constant factor a, written as g(x) = af(x). |
| vertical shift | A vertical translation of a sinusoidal function represented by the additive constant d, which moves the entire graph up or down and shifts the midline. |
| vertical translation | A transformation that shifts the graph of a function up or down by adding a constant k to the function, written as g(x) = f(x) + k. |
| Term | Definition |
|---|---|
| amplitude | The absolute value of the coefficient a in a sinusoidal function, representing the maximum distance from the midline to the peak or trough of the graph. |
| contextual domain | The range of input values for which a sinusoidal function model is meaningful and applicable within a real-world context. |
| frequency | The number of complete cycles of a sinusoidal function that occur over a unit interval of input values. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| periodic phenomena | Events or patterns that repeat at regular intervals over time or space. |
| phase shift | A horizontal translation of a sinusoidal function represented by the constant c, which shifts the graph left or right by -c units. |
| sinusoidal function | A function of the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. |
| sinusoidal regression | A statistical method using technology to fit a sinusoidal function to a data set by estimating the best-fit parameters. |
| vertical shift | A vertical translation of a sinusoidal function represented by the additive constant d, which moves the entire graph up or down and shifts the midline. |
| Term | Definition |
|---|---|
| additive transformation | A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations. |
| asymptote | Lines that a graph approaches but never reaches, indicating behavior at infinity or at points of discontinuity. |
| concave down | A characteristic of a graph where the rate of change is decreasing, creating a curve that opens downward. |
| concave up | A characteristic of a graph where the rate of change is increasing, creating a curve that opens upward. |
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| horizontal dilation | A transformation that stretches or compresses the graph of a function horizontally by multiplying the input by a constant factor b, written as g(x) = f(bx). |
| horizontal translation | A transformation that shifts the graph of a function left or right by adding a constant to the input, written as g(x) = f(x + h). |
| multiplicative transformation | A transformation of a function involving multiplication, resulting in vertical and horizontal dilations. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| periodic | A property of trigonometric functions where they repeat their values at regular intervals. |
| phase shift | A horizontal translation of a sinusoidal function represented by the constant c, which shifts the graph left or right by -c units. |
| point of inflection | A point on the graph of a polynomial where the concavity changes from concave up to concave down or vice versa, occurring where the rate of change changes from increasing to decreasing or decreasing to increasing. |
| reflection over the x-axis | A transformation that flips the graph of a function across the x-axis, occurring when the multiplicative factor is negative in a vertical dilation. |
| reflection over the y-axis | A transformation that flips the graph of a function across the y-axis, occurring when the multiplicative factor is negative in a horizontal dilation. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| slope | The rate of change of a line, representing how much the output changes for each unit change in the input. |
| standard position | The position of an angle with its vertex at the origin and its initial side along the positive x-axis. |
| tangent function | A trigonometric function, denoted f(θ) = tan θ, that gives the slope of the terminal ray of an angle in standard position on the unit circle. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| vertical dilation | A transformation that stretches or compresses the graph of a function vertically by multiplying the function by a constant factor a, written as g(x) = af(x). |
| vertical translation | A transformation that shifts the graph of a function up or down by adding a constant k to the function, written as g(x) = f(x) + k. |
| Term | Definition |
|---|---|
| arccosine | The inverse function of cosine, denoted cos⁻¹(x), that returns an angle whose cosine equals the input value. |
| arcsine | The inverse function of sine, denoted sin⁻¹(x), that returns an angle whose sine equals the input value. |
| arctangent | The inverse function of tangent, denoted tan⁻¹(x), that returns an angle whose tangent equals the input value. |
| inverse trigonometric functions | Functions that reverse the operation of trigonometric functions, taking a trigonometric ratio as input and returning an angle measure as output. |
| invertible function | A function that has an inverse function; a one-to-one function where each output corresponds to exactly one input. |
| periodic | A property of trigonometric functions where they repeat their values at regular intervals. |
| restricted domain | A limited interval of input values for a trigonometric function that makes it one-to-one and therefore invertible. |