The number of times a linear factor (x-c) appears in the factored form of a polynomial, which determines the behavior of the graph at the x-intercept x=c.
If the multiplicity is even, the graph touches the x-axis and turns around; if odd, it crosses the x-axis.
Card 2
Horizontal Asymptote of a Rational Function
Answer
A horizontal line y=L 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.
Students often confuse the rules for horizontal asymptotes (degree comparison) with those for vertical asymptotes (zeros of the denominator).
Card 3
Removable Discontinuity (Hole)
Answer
A point at which a rational function is undefined but where the limit exists, occurring when a common factor (x-c) cancels from both the numerator and denominator.
This is distinct from a vertical asymptote, where the function approaches infinity; at a hole, the graph has a 'gap' that can be filled.
Card 4
Intermediate Value Theorem (IVT)
Answer
For a continuous function f on a closed interval [a, b], if N is any number between f(a) and f(b), then there exists at least one number c in (a, b) such that f(c) = N.
The IVT guarantees the existence of a zero (or any y-value between f(a) and f(b)) if the function is continuous and crosses that y-value.
Card 5
End Behavior of a Polynomial
Answer
The behavior of the graph of a polynomial function as x approaches positive infinity or negative infinity, determined by the degree and leading coefficient of the polynomial.
This describes the direction the graph takes at its far left and far right ends, analogous to horizontal asymptotes for rational functions but usually tending to infinity.
Card 6
Logarithmic Function
Answer
The inverse function of an exponential function, expressed as y = log_b(x) if and only if b^y = x, where b is the base, b > 0, and b 1.
Emphasize the inverse relationship: logarithms answer the question 'to what power must the base be raised to get the argument?'
Card 7
Natural Logarithm
Answer
A logarithm with base e (Euler's number), denoted as ln(x), which is the inverse of the natural exponential function f(x) = e^x.
The 'natural' base e appears frequently in calculus and real-world continuous growth/decay models.
Card 8
Change of Base Formula
Answer
A formula that allows converting a logarithm from one base to another, typically to base 10 or base e: log_b(x) = log_a(x) / log_a(b).
This formula is essential for evaluating logarithms with arbitrary bases using calculators that only have log (base 10) or ln (base e) buttons.
Card 9
Domain of a Logarithmic Function
Answer
For a logarithmic function log_b(g(x)), the domain consists of all x-values for which the argument g(x) is strictly greater than zero.
The argument of a logarithm cannot be zero or negative, which is a common mistake when finding domains.
Card 10
Exponential Growth/Decay Model
Answer
A function of the form P(t) = P_0 * e^(kt) where P_0 is the initial amount, t is time, and k is the continuous growth rate (k>0) or decay rate (k<0).
This specific form using 'e' is crucial for continuous compounding or growth/decay in calculus applications.
Card 11
Radian Measure
Answer
A unit of angular measure where one radian is the measure of the central angle of a circle that subtends an arc equal in length to the radius of the circle.
Radian measure simplifies many trigonometric formulas in calculus, making it the preferred unit for angles in higher mathematics.
Card 12
Period of a Sinusoidal Function
Answer
The length of one complete cycle of a sinusoidal function (sine or cosine), calculated as 2π/|B| for functions of the form A sin(Bx + C) + D or A cos(Bx + C) + D.
The period determines how frequently the graph repeats its pattern, directly affected by the 'B' coefficient.
Card 13
Phase Shift
Answer
The horizontal displacement of a sinusoidal graph from its parent function, calculated as -C/B for functions of the form A sin(Bx + C) + D or A cos(Bx + C) + D.
A common student error is associating (Bx+C) directly with a shift; a positive phase shift value means the graph shifts to the left, while a negative value shifts it to the right.
Card 14
Inverse Trigonometric Function (Arc Function)
Answer
A function that determines the angle corresponding to a given trigonometric ratio, with its domain restricted to ensure it is a one-to-one function and thus has a unique inverse.
The restricted domains (e.g., [-π/2, π/2] for arcsin) are critical for these to be functions and for evaluating unique principal values.
Card 15
Polar Coordinates
Answer
A coordinate system where a point is located by its distance 'r' from the origin (pole) and its angle 'θ' from the positive x-axis (polar axis).
Unlike rectangular coordinates (x,y), a single point can have multiple polar coordinate representations (e.g., by adding 2π to θ or using negative r values).
Card 16
Conversion from Polar to Rectangular Coordinates
Answer
The formulas x = r cos(θ) and y = r sin(θ) used to transform a point from polar coordinates (r, θ) to rectangular coordinates (x, y).
These formulas are derived directly from the definitions of sine and cosine in a right triangle or unit circle.
Card 17
Horizontal Translation (Shift)
Answer
A transformation that moves a graph left or right without changing its shape or orientation, represented by replacing x with (x-h) in the function's equation, where h is the shift amount.
A common error is associating (x-h) with a shift to the left; (x-h) shifts right by h units, and (x+h) shifts left by h units.
Card 18
Vertical Stretch/Compression
Answer
A transformation that multiplies all y-values of a function by a constant factor 'a', stretching the graph vertically if |a| > 1 and compressing it if 0 < |a| < 1.
This transformation affects the amplitude of sinusoidal functions and the steepness of other graphs, applied outside the base function.
Card 19
Reflection Across the x-axis
Answer
A transformation that negates all y-values of a function, resulting in a graph that is a mirror image across the x-axis, represented by y = -f(x).
This is an external transformation, meaning the negative sign is applied to the entire function output.
Card 20
Order of Function Transformations
Answer
The sequence in which multiple transformations are applied to a function's graph, typically following a 'PEMDAS-like' order: reflections/stretches/compressions first, then translations.
Incorrect order of operations (e.g., shifting before stretching) is a frequent source of errors in graphing transformed functions.
Card 21
Slant (Oblique) Asymptote
Answer
A non-horizontal, non-vertical line that the graph of a rational function approaches as x approaches positive or negative infinity, occurring when the degree of the numerator is exactly one greater than the degree of the denominator.
The equation of the slant asymptote is found by performing polynomial long division and taking the quotient (ignoring the remainder).
Card 22
Co-terminal Angles
Answer
Two angles in standard position that have the same initial and terminal sides, differing by an integer multiple of 360 degrees or 2π radians.
Understanding co-terminal angles is essential for finding all solutions to trigonometric equations or simplifying angles outside the [0, 2π) range.
Card 23
Reference Angle
Answer
The acute angle formed by the terminal side of an angle in standard position and the x-axis, always positive and between 0 and π/2 (or 0 and 90 degrees).
Reference angles simplify finding trigonometric values for any angle, as the values are the same as for the reference angle, with signs determined by the quadrant.
Card 24
Unit Circle
Answer
A circle with a radius of 1 unit centered at the origin (0,0) in the Cartesian coordinate system, used to define trigonometric functions for any real number angle.
Points (x,y) on the unit circle directly correspond to (cos θ, sin θ) for the angle θ, making it a fundamental tool for understanding trig values.
Card 25
Period of Tangent/Cotangent
Answer
The length of one complete cycle for the tangent and cotangent functions, which is π radians or 180 degrees, calculated as π/|B| for functions of the form A tan(Bx + C) + D.
Unlike sine and cosine, tangent and cotangent have a period of π, not 2π, due to their repeating nature across vertical asymptotes.
Card 26
Domain of Inverse Tangent (arctan)
Answer
The domain of the arctan(x) function is all real numbers (-∞, ∞), while its range is restricted to (-π/2, π/2) to ensure it is a function.
Students often confuse the domain and range of inverse trig functions; arctan is unique in having an unrestricted domain.
Card 27
Exponential Equation
Answer
An equation in which the variable appears in the exponent, often solved by isolating the exponential term and then taking logarithms of both sides or by expressing both sides with the same base.
The key to solving these is using the inverse relationship between exponentials and logarithms.
Card 28
Logarithmic Equation
Answer
An equation that involves a logarithm of a variable expression, typically solved by using properties of logarithms to condense terms and then converting to exponential form.
Always check solutions against the domain of the original logarithmic terms to avoid extraneous solutions.
Card 29
Polar Equation for a Circle
Answer
A polar equation of the form r = a cos(θ) or r = a sin(θ) represents a circle passing through the origin, or r=k represents a circle centered at the origin.
Understanding these forms helps recognize common shapes in polar coordinates without converting to rectangular form.
Card 30
Vertical Asymptote of a Rational Function
Answer
A vertical line x=c that the graph of a rational function approaches as x approaches c, occurring when the denominator is zero and the numerator is non-zero at x=c.
These are values of x where the function is undefined and tends towards positive or negative infinity, representing an infinite discontinuity.