Glossary
Amplitude
For a periodic function, the distance from the midline to the maximum or minimum value of the graph, representing half the total range.
Example:
In the function y = 3sin(x), the amplitude is 3, meaning the wave oscillates 3 units above and below its midline.
Angles (Unit Circle)
Measurements in radians or degrees on the unit circle, starting from the positive x-axis and rotating counter-clockwise.
Example:
An angle of π/2 radians corresponds to 90 degrees on the unit circle, pointing straight up.
Arccosine (arccos or cos⁻¹)
The inverse cosine function, which returns the angle whose cosine is a given value, typically restricted to the range [0, π].
Example:
When calculating the angle between two vectors, you often use arccosine of their dot product divided by the product of their magnitudes.
Arcsine (arcsin or sin⁻¹)
The inverse sine function, which returns the angle whose sine is a given value, typically restricted to the range [-π/2, π/2].
Example:
If a projectile's vertical displacement divided by its initial velocity is 0.5, you'd use arcsine to find the launch angle.
Arctangent (arctan or tan⁻¹)
The inverse tangent function, which returns the angle whose tangent is a given value, typically restricted to the range (-π/2, π/2).
Example:
To determine the angle of a line on a coordinate plane given its slope, you would apply the arctangent function.
Change of Base Formula
Allows conversion of a logarithm from one base to another: log_b(M) = log_c(M) / log_c(b), useful for calculators.
Example:
To calculate log_7(50) on a calculator, you can use the Change of Base Formula to compute ln(50)/ln(7) or log(50)/log(7).
Circles
A conic section defined as the set of all points equidistant from a fixed center point, with a standard equation (x - h)^2 + (y - k)^2 = r^2.
Example:
The boundary of a circular swimming pool can be described by the equation of a circle.
Component Form (Vectors)
A way to represent a vector using its horizontal and vertical components, typically written as <a, b>.
Example:
A vector representing a displacement of 3 units right and 4 units up would be written in component form as <3, 4>.
Compound Interest
A financial application modeled by A = P(1 + r/n)^(nt), where interest is calculated on the initial principal and also on the accumulated interest from previous periods.
Example:
If you invest $1000 at 5% interest compounded quarterly, the compound interest formula helps calculate your total savings after a few years.
Conic Sections
Curves formed by the intersection of a plane with a double-napped cone, including circles, ellipses, hyperbolas, and parabolas.
Example:
The orbits of planets around the sun are conic sections, specifically ellipses.
Conversion (Polar/Rectangular)
The process of transforming coordinates between the polar system (r, θ) and the rectangular (Cartesian) system (x, y) using formulas like x = r cos θ and y = r sin θ.
Example:
To plot a point given in polar coordinates (4, π/3) on a standard graph, you would use conversion formulas to find its rectangular coordinates (2, 2√3).
Coordinates (Unit Circle)
The (x, y) values of a point on the unit circle corresponding to a given angle, where x = cos θ and y = sin θ.
Example:
For an angle of π/6, the coordinates on the unit circle are (√3/2, 1/2), meaning cos(π/6) = √3/2 and sin(π/6) = 1/2.
Cosecant (csc θ)
The reciprocal of sine, defined as 1 / sin θ.
Example:
If sin θ = 1/2, then cosecant θ = 2.
Cosine (cos θ)
In the context of the unit circle, the x-coordinate of the point corresponding to angle θ; in a right triangle, the ratio of the adjacent side to the hypotenuse.
Example:
The horizontal distance a pendulum swings from its equilibrium position can be modeled using the cosine of its angle.
Cosine (cos)
In a right triangle, the cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Example:
To find the horizontal distance a skateboarder travels down a ramp, you'd use the cosine of the ramp's angle of inclination.
Cotangent (cot θ)
The reciprocal of tangent, defined as 1 / tan θ or cos θ / sin θ.
Example:
If tan θ = 1, then cotangent θ = 1.
Degree
The highest power of the variable in a polynomial function, which indicates the maximum number of roots and influences the graph's end behavior.
Example:
For the polynomial f(x) = 3x^5 - 2x^2 + 7, the degree is 5, telling us it's an odd-degree polynomial.
Determinant (Matrix)
A scalar value calculated from the elements of a square matrix, which provides information about the matrix, such as whether it has an inverse.
Example:
A 2x2 matrix [[a,b],[c,d]] has a determinant of ad - bc, which can tell you if a system of equations has a unique solution.
Domain (Logarithmic)
The set of all possible input values (x-values) for which a logarithmic function is defined, which is typically x > 0.
Example:
The domain of f(x) = log(x-3) is x > 3, because the argument of a logarithm must be positive.
Domain Restriction
The process of limiting the input values (domain) of a function to ensure its inverse is also a function and provides unique outputs.
Example:
For the sine function, a domain restriction to [-π/2, π/2] is necessary so that arcsin produces a unique angle for each ratio.
Dot Product
An operation between two vectors that results in a scalar quantity, calculated as the sum of the products of their corresponding components (a₁b₁ + a₂b₂).
Example:
The dot product of <1, 2> and <3, -1> is (13) + (2-1) = 3 - 2 = 1, which can be used to find the angle between vectors.
Double Angle Identities
Formulas used to express trigonometric functions of twice an angle in terms of functions of the angle itself, e.g., sin(2θ).
Example:
The Double Angle Identity for sine, sin(2θ) = 2sinθcosθ, is useful for simplifying expressions or solving equations.
Eliminating the Parameter
The process of converting a set of parametric equations into a single Cartesian (rectangular) equation by solving one parametric equation for the parameter and substituting it into the other.
Example:
Given x = t+1 and y = t^2, eliminating the parameter 't' yields y = (x-1)^2, which is a parabola.
Ellipses
A conic section defined as the set of all points where the sum of the distances from two fixed points (foci) is constant, with a standard equation involving a^2 and b^2.
Example:
The shape of a stadium track or a planetary orbit is often an ellipse.
End Behavior
Describes what happens to the function's output (y-values) as the input (x-values) approaches positive or negative infinity.
Example:
A polynomial like f(x) = x^3 has end behavior where as x approaches positive infinity, f(x) approaches positive infinity, and as x approaches negative infinity, f(x) approaches negative infinity.
Exponential Function
A function of the form f(x) = a * b^x, where 'a' is the initial value, 'b' is the base (a positive number not equal to 1), and 'x' is the exponent.
Example:
Population growth often follows an exponential function, like P(t) = 100 * (1.05)^t, where the population increases by a fixed percentage over time.
Growth and Decay (Continuous)
Models for continuous exponential change, represented by A = A₀e^(kt), where A₀ is the initial amount, k is the continuous growth/decay rate, and t is time.
Example:
Radioactive decay of a substance is often modeled using continuous growth and decay, where 'k' would be a negative value.
Growth/Decay
Describes the behavior of an exponential function: growth occurs when the base 'b' is greater than 1, and decay occurs when 'b' is between 0 and 1.
Example:
A car's value depreciating by 15% each year is an example of exponential decay, while compound interest is exponential growth.
Holes
Points of discontinuity in the graph of a rational function that occur when a common factor cancels out from both the numerator and the denominator.
Example:
In f(x) = (x^2-4)/(x-2), there is a hole at x=2 because the factor (x-2) cancels out, leaving a removable discontinuity.
Horizontal Asymptote (Exponential)
A horizontal line that an exponential function's graph approaches as x tends towards positive or negative infinity, typically y=0 unless shifted.
Example:
The function f(x) = 2^x has a horizontal asymptote at y=0, meaning the graph gets infinitely close to the x-axis as x approaches negative infinity.
Horizontal Asymptotes
Horizontal lines that the graph of a rational function approaches as x approaches positive or negative infinity, determined by comparing the degrees of the numerator and denominator.
Example:
The function f(x) = (2x+1)/(x-3) has a horizontal asymptote at y=2, meaning the graph flattens out at this y-value for very large or small x.
Hyperbolas
A conic section defined as the set of all points where the absolute difference of the distances from two fixed points (foci) is constant, forming two separate branches.
Example:
The path of a comet that does not orbit the sun, but rather passes by once, can be a hyperbola.
Inverse (Matrix)
For a square matrix A, its inverse (A⁻¹) is another matrix such that when multiplied by A, it results in the identity matrix.
Example:
To 'undo' a linear transformation represented by a matrix, you would apply its inverse matrix.
Inverse Trigonometric Functions
Functions that determine the angle corresponding to a given trigonometric ratio. They are the inverses of sine, cosine, and tangent.
Example:
If you know the ratio of a ramp's height to its length, inverse trigonometric functions can help you find the angle of the ramp's incline.
Leading Coefficient
The coefficient of the term with the highest power in a polynomial function, which, along with the degree, determines the end behavior.
Example:
In f(x) = -2x^4 + 5x - 1, the leading coefficient is -2, indicating that both ends of the graph will go downwards.
Linear Transformations
Functions that map vectors from one vector space to another, preserving vector addition and scalar multiplication, often represented by matrix multiplication.
Example:
Rotating an object in a 2D plane is a linear transformation that can be achieved by multiplying its coordinate vectors by a rotation matrix.
Logarithmic Function
The inverse of an exponential function, written as y = log_b(x), which means b^y = x.
Example:
The Richter scale, used to measure earthquake intensity, is based on a logarithmic function, where each whole number increase represents a tenfold increase in amplitude.
Magnitude (Vector)
The length or size of a vector, calculated using the Pythagorean theorem: |a| = √(a₁^2 + a₂^2).
Example:
The magnitude of the velocity vector <3, 4> is √(3^2 + 4^2) = 5, representing the speed.
Matrices
Rectangular arrays of numbers, symbols, or expressions arranged in rows and columns, used to organize data and perform linear transformations.
Example:
A spreadsheet can be thought of as a matrix where each cell contains a number.
Matrix Addition/Subtraction
Operations performed by adding or subtracting corresponding elements of two matrices, which must have the same dimensions.
Example:
To combine two sales reports, each represented as a matrix, you would use matrix addition to sum the corresponding entries.
Matrix Multiplication
A more complex operation where rows of the first matrix are multiplied by columns of the second matrix, requiring compatible dimensions (inner dimensions must match).
Example:
In computer graphics, applying a rotation to an image involves matrix multiplication of the image's coordinate matrix by a rotation matrix.
Matrix Scalar Multiplication
The operation of multiplying every element of a matrix by a single real number (scalar).
Example:
To double all values in a matrix, you would perform matrix scalar multiplication by 2.
Multiplicity
The number of times a root appears in a polynomial's factorization, determining whether the graph crosses or touches the x-axis at that zero.
Example:
If (x-3)^2 is a factor, the root x=3 has a multiplicity of 2, meaning the graph will touch the x-axis at 3 and turn around.
Parabolas
A conic section defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix), with a standard equation involving 4p.
Example:
The trajectory of a projectile under gravity, or the shape of a satellite dish, is a parabola.
Parametric Equations
A set of equations that define the coordinates of points on a curve as functions of a third variable, called a parameter (often 't').
Example:
The position of a moving object over time can be described by parametric equations like x(t) = 2t and y(t) = t^2, where 't' is time.
Period (Trigonometric)
The horizontal length of one complete cycle of a periodic function before its graph begins to repeat.
Example:
The period of y = sin(2x) is π, meaning the graph completes one full wave in a horizontal distance of π units.
Periodic Function
A function that repeats its values in regular intervals or periods. Trigonometric functions are periodic.
Example:
The motion of a Ferris wheel is modeled by a periodic function because its height repeats every full rotation.
Periodic Functions
Functions whose graphs repeat their pattern over a regular interval, known as the period.
Example:
The height of a point on a Ferris wheel as it rotates is an example of a periodic function.
Phase Shift
A horizontal translation of a periodic function's graph, indicating how much the graph is shifted left or right from its standard position.
Example:
The function y = sin(x - π/4) has a phase shift of π/4 units to the right compared to y = sin(x).
Polar Coordinates
A coordinate system that represents points in a plane using a distance 'r' from the origin and an angle 'θ' from the positive x-axis.
Example:
Instead of (3,4) in Cartesian, a point might be described by its distance from the origin and its angle, like (5, 53.1°) in polar coordinates.
Polynomial Function
A function that can be written in the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where 'n' is a non-negative integer and coefficients are real numbers.
Example:
The path of a thrown ball can be modeled by a polynomial function like f(x) = -0.5x^2 + 2x + 3, showing its parabolic trajectory.
Power Rule (Logarithms)
A property stating that the logarithm of a number raised to a power is the power times the logarithm of the number: log_b(M^p) = p log_b(M).
Example:
To simplify log(x^4), you can use the Power Rule to write it as 4 log(x).
Product Rule (Logarithms)
A property stating that the logarithm of a product is the sum of the logarithms: log_b(MN) = log_b(M) + log_b(N).
Example:
Using the Product Rule, log(5x) can be expanded to log(5) + log(x), simplifying expressions.
Pythagorean Identities
Fundamental trigonometric identities derived from the Pythagorean theorem on the unit circle, such as sin^2 θ + cos^2 θ = 1.
Example:
When simplifying a trigonometric expression, you can often substitute 1 for sin^2 θ + cos^2 θ using the Pythagorean Identity.
Pythagorean Theorem
A fundamental theorem in geometry stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Example:
If you know two sides of a right triangle, you can use the Pythagorean Theorem to find the length of the third side, which is useful in inverse trig problems.
Quotient Rule (Logarithms)
A property stating that the logarithm of a quotient is the difference of the logarithms: log_b(M/N) = log_b(M) - log_b(N).
Example:
The expression log_3(x/y) can be rewritten as log_3(x) - log_3(y) using the Quotient Rule.
Range
The set of all possible output values of a function. For inverse trigonometric functions, the range represents the specific angles they can return.
Example:
The range of arccosine is [0, π], meaning it will only output angles in the first or second quadrant.
Range (Logarithmic)
The set of all possible output values (y-values) for a logarithmic function, which is always all real numbers.
Example:
No matter the base or horizontal shifts, the range of a logarithmic function will always span from negative infinity to positive infinity.
Rational Function
A function that can be expressed as the ratio of two polynomials, f(x) = P(x)/Q(x), where Q(x) is not the zero polynomial.
Example:
The cost per item when producing a large quantity can often be modeled by a rational function, showing how average cost changes with volume.
Scalar Multiplication (Vectors)
The operation of multiplying a vector by a scalar (a real number), which scales the vector's magnitude and can reverse its direction.
Example:
Multiplying the vector <2, 3> by the scalar 5 results in <10, 15>, demonstrating scalar multiplication.
Secant (sec θ)
The reciprocal of cosine, defined as 1 / cos θ.
Example:
When cos θ = √2/2, then secant θ = 2/√2 = √2.
Simple Harmonic Motion
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to the displacement, often modeled by sine or cosine functions.
Example:
The oscillation of a mass on a spring or the swing of a pendulum (for small angles) are classic examples of simple harmonic motion.
Sine (sin θ)
In the context of the unit circle, the y-coordinate of the point corresponding to angle θ; in a right triangle, the ratio of the opposite side to the hypotenuse.
Example:
If a ladder leans against a wall at a 60-degree angle, the height it reaches on the wall is related to the sine of that angle.
Sine (sin)
In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
Example:
If a ladder leans against a wall, the sine of the angle it makes with the ground helps determine how high up the wall it reaches relative to its length.
Sum and Difference Identities
Formulas used to find the sine, cosine, or tangent of the sum or difference of two angles, e.g., sin(A ± B).
Example:
To find the exact value of cos(75°), you can use the Sum and Difference Identity for cosine: cos(45° + 30°).
Systems of Equations (Matrices)
A set of two or more equations with the same variables, which can be solved efficiently using matrix operations like finding the inverse or row reduction.
Example:
Solving for the intersection point of two lines, like 2x+y=5 and x+y=3, can be done by representing them as a system of equations and using matrices.
Tangent (tan θ)
The ratio of sine to cosine (sin θ / cos θ); in a right triangle, the ratio of the opposite side to the adjacent side.
Example:
The slope of a line can be found using the tangent of the angle the line makes with the positive x-axis.
Tangent (tan)
In a right triangle, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Example:
Surveyors use the tangent of an angle to calculate the height of a tall building by measuring the distance from its base and the angle of elevation.
Transformations
Changes applied to a function's graph, such as shifts (horizontal/vertical), stretches/compressions, and reflections, which alter its position or shape.
Example:
Applying a vertical shift of +3 to f(x) = x^2 results in g(x) = x^2 + 3, moving the entire parabola up by 3 units, a common transformation.
Trigonometric Functions
Functions that relate angles of a right triangle to the ratios of its side lengths. They are fundamental for analyzing periodic phenomena.
Example:
When designing a roller coaster, engineers use trigonometric functions to calculate the height of a car at different points along a curved track.
Unit Circle
A circle with a radius of 1 centered at the origin, used as a fundamental tool to define trigonometric functions for any angle.
Example:
To find the sine and cosine of 90 degrees, you can locate the point (0,1) on the unit circle.
Unit Circle
A circle with a radius of one unit centered at the origin of a coordinate plane, used to visualize trigonometric values for various angles.
Example:
To quickly evaluate sin(π/6) or cos(π/3), students often refer to the coordinates on the unit circle.
Unit Vectors
Vectors with a magnitude of 1, often used to indicate direction, such as 'i' for the x-direction and 'j' for the y-direction.
Example:
The vector <0, 1> is a unit vector pointing directly upwards along the y-axis.
Vector Addition/Subtraction
Operations performed by adding or subtracting the corresponding components of two or more vectors.
Example:
If you walk 3 miles east (<3,0>) and then 4 miles north (<0,4>), your total displacement is found by vector addition: <3,4>.
Vectors
Quantities that possess both magnitude (length) and direction, often represented as arrows or in component form.
Example:
When describing the velocity of an airplane, you need both its speed (magnitude) and its direction of travel, making velocity a vector quantity.
Vertical Asymptote (Logarithmic)
A vertical line that a logarithmic function's graph approaches but never touches, typically x=0 unless shifted horizontally.
Example:
The function f(x) = log(x) has a vertical asymptote at x=0, meaning the graph gets infinitely close to the y-axis as x approaches 0 from the right.
Vertical Asymptotes
Vertical lines that the graph of a rational function approaches but never touches, occurring where the denominator is zero and the numerator is not.
Example:
For f(x) = 1/(x-2), there is a vertical asymptote at x=2, indicating the function's value becomes infinitely large or small as x approaches 2.
Vertical Shift
A vertical translation of a function's graph, moving the entire graph up or down.
Example:
Adding a constant 'D' to a sine function, like y = sin(x) + 5, results in a vertical shift of 5 units upwards, moving the midline.
X-intercept (Logarithmic)
The point where a logarithmic function's graph crosses the x-axis, which is typically (1, 0) for a basic logarithmic function.
Example:
For the function f(x) = log_2(x), the x-intercept is (1, 0), as 2^0 = 1.
Y-intercept (Exponential)
The point where an exponential function's graph crosses the y-axis, which is the initial value 'a' in the form f(x) = a * b^x.
Example:
For the function f(x) = 5 * (1.2)^x, the y-intercept is (0, 5), representing the starting amount.
Zeros (Roots)
The x-values where a function equals zero, corresponding to the points where the graph intersects the x-axis.
Example:
If a quadratic function models the height of a rocket, its positive zero would represent the time it hits the ground.
pH Scale
A logarithmic scale used to specify the acidity or basicity of an aqueous solution, defined as pH = -log[H+], where [H+] is the hydrogen ion concentration.
Example:
Lemon juice has a low pH, indicating a high hydrogen ion concentration, which is measured using the pH scale.