AP Precalculus Vocabulary

30-60-90 Triangle

A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90° whose sides are always in the ratio 1 : √3 : 2, where the shortest side is opposite the 30° angle and the hypotenuse is twice the shortest side.

Example: "In a          with hypotenuse 10, the side opposite the 30° angle is 5 and the side opposite the 60° angle is 5√3, following the ratio 1 : √3 : 2."

45-45-90 Triangle

A 45-45-90 triangle is a special right triangle with two equal acute angles of 45° whose sides are always in the ratio 1 : 1 : √2, meaning the hypotenuse equals the length of either leg multiplied by √2.

Example: "In a          with legs of length 7, the hypotenuse is 7√2, since the side ratios are always 1 : 1 : √2."

Abscissa

The abscissa is the x-coordinate (first value) in an ordered pair. For the point (8, –2), the abscissa is 8. See also: ordinate.

Example: "For the point (−3, 7) on the graph of y = x² + 2x + 10, the          is −3, which is the input value at which the function is evaluated."

Absolute Maximum

Absolute maximum is the highest point of a function across its entire domain. Learn how to find it using derivative tests.

Example: "The function f(x) = −x² + 4x − 1 has an          of 3 at x = 2, since the parabola opens downward and the vertex represents the highest point over all real numbers."

Absolute Minimum

Absolute minimum is the lowest point of a function over its entire domain. Also called global minimum. Includes related tests and terms.

Example: "On the closed interval [0, 4], the function f(x) = x² − 4x + 5 achieves its          value of 1 at x = 2, the vertex of the parabola."

Absolute Value

Absolute value is a number's distance from zero. |–6| = 6 and |8| = 8. Negative numbers become positive; zero stays zero.

Example: "To solve |2x − 5| = 9, we recognize that the          represents distance from zero and split it into two cases: 2x − 5 = 9 and 2x − 5 = −9, giving x = 7 or x = −2."

Absolute Value Equation

An absolute value equation is an equation containing an absolute value expression, solved by splitting it into two cases: one where the expression inside is positive and one where it is negative.

Example: "The          |3x + 1| = 10 yields two linear equations, 3x + 1 = 10 and 3x + 1 = −10, whose solutions are x = 3 and x = −11/3."

Absolute Value Function

The absolute value function f(x) = |x| returns the non-negative distance of x from zero, producing a V-shaped graph with vertex at the origin that is decreasing for x < 0 and increasing for x > 0.

Example: "The          f(x) = |x − 2| + 3 has a vertex at (2, 3) and is decreasing on (−∞, 2) and increasing on (2, ∞), forming a V-shape on the coordinate plane."

Absolute Value Graph

The graph of an absolute value function is V-shaped, with the vertex occurring where the expression inside the absolute value equals zero. Transformations shift, reflect, or stretch the V while preserving its characteristic shape.

Example: "The          of y = |x + 4| − 1 is a V-shaped curve with vertex at (−4, −1), where the left branch has slope −1 and the right branch has slope 1."

Absolute Value Inequality

An absolute value inequality is an inequality involving an absolute value expression. The form |x| < c yields a compound inequality −c < x < c, while |x| > c yields x < −c or x > c.

Example: "Solving the          |x − 3| < 5 gives −2 < x < 8, since we require x to be within 5 units of 3 on the number line."

Absolute Value of Complex Number

Absolute value is a number's distance from zero. |–6| = 6 and |8| = 8. Negative numbers become positive; zero stays zero.

Example: "The absolute value of the complex number 3 + 4i is √(3² + 4²) = √25 = 5, which represents its distance from the origin in the complex plane."

Acute Angle

An acute angle is any angle measuring less than 90°. Explore examples, diagrams, and related terms like obtuse angle to master angle types.

Example: "In the unit circle, a terminal ray in the first quadrant forms an          with the positive x-axis, so both sine and cosine are positive for any such angle."

Adjacent Angles

Adjacent angles share a common vertex and side without overlapping. Understand this fundamental geometry concept with clear examples.

Example: "When two          together form a straight angle of 180°, they are supplementary; if one measures 65°, the other must measure 115°."

Ambiguous Case

The ambiguous case (SSA) arises in triangle solving when two sides and a non-included angle are given, potentially producing zero, one, or two valid triangles depending on the relative lengths of the sides.

Example: "When applying the Law of Sines with two sides and a non-included angle (SSA), the          may arise, potentially yielding two distinct triangles, one triangle, or no triangle at all."

Amplitude

Amplitude is half the difference between the minimum and maximum values of a periodic function's range — essentially the radius of the range.

Example: "For f(x) = 3 sin(2x), the          is 3, meaning the function oscillates between −3 and 3, with the maximum value occurring at x = π/4."

Amplitude of Sine

Amplitude is half the difference between the minimum and maximum values of a periodic function's range — essentially the radius of the range.

Example: "The          in y = −4 sin(x) is 4, since amplitude is always taken as the absolute value of the leading coefficient, indicating the function reaches a maximum of 4 and a minimum of −4."

Angle

An angle is formed by two rays sharing a common endpoint, measured in degrees or radians. Explore related terms like arm, initial side, and terminal side.

Example: "An          measuring 5π/6 radians is formed when the terminal ray rotates counterclockwise from the positive x-axis, landing in the second quadrant at 150°."

Angle Between Vectors

The angle between two vectors u and v is found using the formula cos θ = (u · v) / (|u| |v|), where u · v is their dot product and |u|, |v| are their magnitudes. The result θ is always in the range [0, π].

Example: "To find the          u = ⟨1, 2⟩ and v = ⟨3, 4⟩, we use cos θ = (u · v) / (|u| |v|) = 11 / (√5 · √25) = 11 / (5√5), then apply arccos."

Angle in Standard Position

Standard position is an angle on the x-y plane with its starting side on the positive x-axis, turning counterclockwise.

Example: "An angle of 240° in standard position has its initial side on the positive x-axis and its terminal side in the third quadrant, making it a reflex-equivalent angle with reference angle 60°."

Angle of Depression

The angle of depression is the angle measured downward from the horizontal to the line of sight toward an object below the observer. It equals the angle of elevation from the lower object to the observer.

Example: "From the top of a 50-meter cliff, an observer sights a boat with an          of 30°, allowing the horizontal distance to be calculated as 50 / tan(30°) = 50√3 meters."

Angle of Elevation

The angle of elevation is the angle measured upward from the horizontal to the line of sight toward an object above the observer. It is used with right triangle trigonometry to find heights and distances.

Example: "Standing 100 feet from a building and measuring an          of 60° to the top, we can find the height using tan(60°) = h/100, giving h = 100√3 feet."

Angular Velocity

Angular velocity ω is the rate at which an angle changes with respect to time, measured in radians per second. It relates to linear velocity v and radius r by the formula v = rω.

Example: "A wheel completing 4 full revolutions per second has an          of ω = 4 · 2π = 8π radians per second, which can be used to find the linear speed of a point on its rim."

Arc Length

Arc length is the distance along a curved portion of a circle, calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians.

Example: "A sector with radius 6 and central angle 2π/3 radians has          s = rθ = 6 · (2π/3) = 4π, which is the distance along the curved portion of the sector."

Arccos

Arccos (inverse cosine) is the function that returns the angle whose cosine equals a given value. Its domain is [−1, 1] and its range is restricted to [0, π] to ensure a unique output.

Example: "Since cos(π/3) = 1/2, we have         (1/2) = π/3, where          is the inverse cosine function restricted to the range [0, π] to ensure a unique output."

Archimedean Spiral

An Archimedean spiral is a polar curve of the form r = aθ in which the distance between successive loops is constant. Unlike logarithmic spirals, the spacing does not grow exponentially as the angle increases.

Example: "The          r = 2θ expands outward at a constant rate as θ increases, unlike exponential spirals, because each successive loop is equally spaced from the previous one."

Arcsin

Arcsin (inverse sine) is the function that returns the angle whose sine equals a given value. Its domain is [−1, 1] and its range is restricted to [−π/2, π/2] to ensure a unique output.

Example: "Because sin(π/6) = 1/2, we evaluate         (1/2) = π/6, noting that the          function is restricted to the range [−π/2, π/2] to maintain a one-to-one relationship."

Arctan

Arctan (inverse tangent) is the function that returns the angle whose tangent equals a given value. Its domain is all real numbers and its range is restricted to (−π/2, π/2), with horizontal asymptotes at ±π/2.

Example: "The expression         (1) = π/4, since tan(π/4) = 1 and the range of          is (−π/2, π/2), making it the principal value of the inverse tangent."

Area of a Triangle

The area of a triangle can be found using the formula Area = (1/2)ab sin C, where a and b are two sides and C is the included angle between them. This generalizes the standard base-times-height formula to non-right triangles.

Example: "Using the formula Area = (1/2)ab sin C, a triangle with sides a = 8, b = 5, and included angle C = 30° has area (1/2)(8)(5) sin(30°) = 10 square units."

Argument of Complex Number

The argument of a complex number a + bi is the angle θ that the number makes with the positive real axis in the complex plane, calculated as θ = arctan(b/a) with quadrant adjustments. It is used in the polar form r(cos θ + i sin θ).

Example: "The complex number −1 + i√3 has argument θ = arctan(√3 / −1) + π = 2π/3, placing it in the second quadrant of the complex plane with modulus 2."

Arithmetic Mean

Arithmetic mean is the most common type of average, found by adding numbers and dividing by how many numbers there are.

Example: "The          of the values 7, 13, 19, and 25 is (7 + 13 + 19 + 25) / 4 = 16, which also happens to be the middle value of this equally spaced arithmetic sequence."

Arithmetic Sequence

An arithmetic sequence has a constant difference between terms. Find the explicit formula aₙ = a₁ + (n–1)d with examples and key vocabulary.

Example: "The          5, 11, 17, 23, … has a common difference of 6, so the explicit formula is aₙ = 5 + (n − 1) · 6 = 6n − 1, giving a₁₀ = 59."

Arithmetic Series

An arithmetic series is the sum of terms with a constant difference. Find the sum using the first term, last term, and number of terms.

Example: "The sum of the          3 + 7 + 11 + … + 99 can be found using S = n(a₁ + aₙ)/2 = 25(3 + 99)/2 = 1275, where there are 25 terms with common difference 4."

Associative Property

The associative property states that the grouping of numbers in addition or multiplication does not affect the result: (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c). This property does not hold for subtraction or division.

Example: "When simplifying (2 · 3) · 5 = 2 · (3 · 5) = 30, we apply the          of multiplication, which confirms that regrouping factors does not change the product."

Asymptote

A line or curve a graph approaches but never crosses (vertically). Understand asymptotes—horizontal, vertical, and oblique—explained clearly.

Example: "The rational function f(x) = (2x + 1)/(x − 3) has a vertical          at x = 3 and a horizontal          at y = 2, which the graph approaches but never crosses."

Asymptotes of Hyperbola

The asymptotes of a hyperbola are the two lines that the branches of the hyperbola approach but never reach. For a hyperbola in standard form x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x.

Example: "The hyperbola x²/9 − y²/4 = 1 has asymptotes y = ±(2/3)x, which are the diagonal lines the hyperbola's branches approach as x tends toward ±∞."

Augmented Matrix

An augmented matrix combines a system's coefficient matrix with its constants column, separated by a vertical line, to represent a linear system in matrix form.

Example: "The system 2x + y = 5 and x − 3y = −4 can be written as the          [[2, 1 | 5], [1, −3 | −4]], which is then row-reduced to find the solution."

Average Rate of Change

The average rate of change of a function f over an interval [a, b] is the ratio (f(b) − f(a))/(b − a), representing the slope of the secant line connecting the two endpoints of the graph on that interval.

Example: "The          of f(x) = x³ on [1, 4] is (f(4) − f(1))/(4 − 1) = (64 − 1)/3 = 21, representing the slope of the secant line connecting the two endpoints."

Average Rate of Change Formula

The average rate of change formula is [f(b) − f(a)] / (b − a), which computes the slope of the secant line through (a, f(a)) and (b, f(b)). It measures how much the output changes per unit change in input over the interval.

Example: "Applying the          [f(b) − f(a)] / (b − a) to g(x) = √x over [4, 9] gives (3 − 2)/(9 − 4) = 1/5, the slope of the secant line."

Axis of Symmetry

The axis of symmetry of a parabola is the vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For y = ax² + bx + c, it is the line x = −b/(2a).

Example: "For the parabola y = 2x² − 8x + 3, the          is x = −(−8)/(2 · 2) = 2, and the vertex lies on this vertical line at the point (2, −5)."

Bearing

Bearing refers to two methods of indicating direction using compass points or angles measured clockwise from north.

Example: "A ship traveling at a          of 135° is heading southeast, 135° clockwise from due north, and this direction can be converted to a standard trigonometric angle for navigation calculations."

Biconditional

A biconditional statement (written P ↔ Q, or "P if and only if Q") is true when both P and Q have the same truth value. It asserts that the conditional P → Q and its converse Q → P are both true simultaneously.

Example: "The statement 'A function has an inverse if and only if it is one-to-one' is a         , meaning both the original conditional and its converse are simultaneously true."

Bijective Function

A bijective function is one that is both injective (one-to-one) and surjective (onto), meaning every output corresponds to exactly one input and every element of the codomain is reached. Bijective functions always have an inverse.

Example: "The function f(x) = 2x + 1 defined from ℝ to ℝ is bijective because it is both injective (each output comes from a unique input) and surjective (every real number is an output), guaranteeing an inverse function."

Binomial

A binomial is a polynomial with exactly two unlike terms, like 2x – 3 or 3x⁵ + 8x⁴. See examples and related terms like monomial and trinomial.

Example: "The          x + 3 appears in the factored form of x² + 6x + 9 = (x + 3)², illustrating how recognizing a          factor simplifies polynomial expressions."

Binomial Theorem

Binomial Theorem explains how to expand powers of binomials using formulas and Pascal's Triangle coefficients.

Example: "Using the         , (x + 2)⁴ = x⁴ + 4·2x³ + 6·4x² + 4·8x + 16 = x⁴ + 8x³ + 24x² + 32x + 16, where coefficients come from row 4 of Pascal's Triangle."

Bounded Function

A bounded function is one whose output values are confined within a finite range, meaning there exist real numbers m and M such that m ≤ f(x) ≤ M for all x in its domain. Sine and cosine are classic examples.

Example: "The sine function is a          because for all real x, −1 ≤ sin(x) ≤ 1, meaning its range is confined within a finite interval regardless of the input."

Box Method

The box method is a visual technique for multiplying polynomials by organizing the partial products in a rectangular grid, with one polynomial's terms along the top and the other's along the side. The products are then summed to obtain the result.

Example: "Using the          to multiply (2x + 3)(x − 5), we place terms along the outside and fill in four products—2x², −10x, 3x, −15—then combine like terms to get 2x² − 7x − 15."

Cardioid

A heart-shaped curve traced by a point on a rolling circle. Learn cardioid equations in polar coordinates and how it relates to other curves.

Example: "The polar curve r = 1 + cos θ is a          that passes through the origin when θ = π and reaches its maximum radius of 2 when θ = 0, tracing a heart-shaped path."

Ceiling Function

Ceiling function rounds any number up to the nearest integer. Explore the least integer function, notation, examples, and its relation to the floor function.

Example: "The          ⌈2.3⌉ = 3, and ⌈−1.7⌉ = −1, since the ceiling always rounds up to the nearest integer regardless of whether the number is positive or negative."

Center of a Circle

The center of a circle is the fixed point equidistant from all points on the circle. In the standard form equation (x − h)² + (y − k)² = r², the center is at (h, k).

Example: "For the circle defined by (x − 2)² + (y + 5)² = 49, the center of the circle is at (2, −5) and the radius is 7, as read directly from the standard form equation."

Center of Ellipse

The center of an ellipse is the midpoint between the two foci and between the two vertices. In the standard form (x − h)²/a² + (y − k)²/b² = 1, the center is at (h, k).

Example: "The ellipse (x + 1)²/16 + (y − 3)²/9 = 1 has its          at (−1, 3), with a semi-major axis of 4 along the horizontal direction and semi-minor axis of 3 vertically."

Central Angle

A central angle is an angle inside a circle whose vertex is at the center. Explore its relationship to arcs and more circle concepts.

Example: "A          of π/3 radians in a circle of radius 9 subtends an arc of length rθ = 9 · π/3 = 3π, connecting arc length, radius, and the          via the formula s = rθ."

Circle

A circle is the set of all points a fixed distance from a center point. Explore related concepts like area, circumference, and radius.

Example: "The          with equation x² + y² = 25 has its center at the origin and a radius of 5, representing all points exactly 5 units from (0, 0) in the coordinate plane."

Circle in Polar Form

A circle is the set of all points a fixed distance from a center point. Explore related concepts like area, circumference, and radius.

Example: "In polar coordinates, the circle r = 4 sin θ has center at the Cartesian point (0, 2) and radius 2, demonstrating how some circles have elegant representations in polar form."

Closed Interval

A closed interval is an interval that includes both of its endpoints. Explore related concepts like open, half-open, and half-closed intervals.

Example: "The domain of f(x) = √(4 − x²) is the          [−2, 2], meaning both endpoints x = −2 and x = 2 are included since the radicand equals zero there."

Co-vertices

The co-vertices of an ellipse are the endpoints of the minor axis, located at a distance of b (the semi-minor axis length) from the center perpendicular to the major axis.

Example: "For the ellipse x²/25 + y²/9 = 1, the          are at (0, 3) and (0, −3), located on the minor axis, while the vertices at (±5, 0) lie on the major axis."

Coefficient

A coefficient is the number multiplied by variables in a term. For example, 123 is the coefficient in 123x³y. Perfect for math students and homework help.

Example: "In the polynomial 5x³ − 3x² + 7x − 2, the          of the x² term is −3, which determines the contribution of that degree-2 term to the function's behavior."

Coefficient Matrix

A coefficient matrix is the matrix formed by the coefficients in a linear system of equations. Understand how it differs from an augmented matrix.

Example: "The system 3x − y + 2z = 7, x + 4y − z = 1, and 2x + y + 5z = 9 has          [[3, −1, 2], [1, 4, −1], [2, 1, 5]], which can be used to compute the determinant and check for a unique solution."

Cofactor

A cofactor is a matrix determinant found by deleting one element's row and column, then applying a + or – sign based on the element's position.

Example: "To expand the determinant of a 3×3 matrix along the first row, each entry is multiplied by its         —the determinant of the 2×2 minor obtained by deleting that entry's row and column, signed by (−1)^(i+j)."

Cofunction Identity

A cofunction identity expresses one trigonometric function in terms of its complementary counterpart, such as sin(θ) = cos(π/2 − θ) and tan(θ) = cot(π/2 − θ). These identities reflect the relationship between complementary angles in a right triangle.

Example: "Using the          sin(θ) = cos(π/2 − θ), we can rewrite sin(70°) = cos(20°), which is useful when simplifying expressions involving complementary angles."

Cofunction Theorem

The cofunction theorem states that any trigonometric function of an acute angle equals the cofunction of its complement: sin(θ) = cos(90° − θ), tan(θ) = cot(90° − θ), and sec(θ) = csc(90° − θ).

Example: "The          states that sin(40°) = cos(50°) because 40° and 50° are complementary angles, so the sine and cosine of complementary angles are always equal."

Combination

Combination: a selection of objects where order doesn't matter. Learn how combinations differ from permutations with clear examples and formulas.

Example: "When choosing a committee of 3 students from a class of 12, the number of ways is C(12, 3) = 12! / (3! · 9!) = 220, since the order in which members are selected does not matter."

Common Difference

The common difference is the constant value added to each term of an arithmetic sequence to produce the next term. If aₙ is the nth term, the common difference d = aₙ − aₙ₋₁.

Example: "The sequence 7, 12, 17, 22, … is arithmetic with a          of 5, and this constant difference means the explicit formula is aₙ = 7 + (n − 1) · 5 = 5n + 2."

Common Ratio

Common ratio is the fixed multiplier between terms in a geometric sequence. For 3, 6, 12, 24… the common ratio r = 2.

Example: "In the geometric sequence 2, 6, 18, 54, …, the          is r = 3, so the nth term is given by aₙ = 2 · 3^(n−1) and the sequence grows exponentially."

Commutative Property

Commutative means a+b=b+a for all a and b. Addition and multiplication are commutative; subtraction and division are not.

Example: "When computing the dot product u · v = ⟨2, 5⟩ · ⟨3, 1⟩ = 11, we can verify the          holds since v · u = ⟨3, 1⟩ · ⟨2, 5⟩ = 11 as well."

Complement

The complement of a set A (written A') contains all elements in the universal set that are not in A. In probability, P(A') = 1 − P(A), since all outcomes must belong to either A or its complement.

Example: "If the probability that event A occurs is 0.35, then the probability of its          A' (event A does not occur) is 1 − 0.35 = 0.65, since all probabilities must sum to 1."

Complementary

Two angles are complementary if their measures sum to 90°. In a right triangle, the two acute angles are always complementary, which is why the cofunction identities relate sine and cosine of complementary angles.

Example: "The angles 34° and 56° are          because 34° + 56° = 90°, which means sin(34°) = cos(56°) by the cofunction relationship between sine and cosine."

Complementary Angles

Complementary angles are two angles whose measures add up to exactly 90°. They frequently appear in right triangle trigonometry and are the basis for cofunction identities such as sin(θ) = cos(90° − θ).

Example: "In a right triangle, the two acute angles are always          summing to 90°, which explains why sin of one equals cos of the other."

Completing the Square

Completing the square is an algebraic technique for rewriting a quadratic expression ax² + bx + c in the form a(x − h)² + k by adding and subtracting a strategic constant. It is used to convert to vertex form and to derive the quadratic formula.

Example: "By          on x² + 6x + 2, we rewrite it as (x + 3)² − 7, revealing that the vertex of the corresponding parabola is at (−3, −7)."

Complex Arithmetic

Complex arithmetic involves addition, subtraction, multiplication, and division of complex numbers using the rule i² = −1. Real parts combine with real parts and imaginary parts combine with imaginary parts during addition and subtraction.

Example: "Performing         , (3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i, using the fact that i² = −1."

Complex Conjugate

Complex conjugate of a + bi is a – bi, changing only the imaginary part's sign while keeping the real part unchanged.

Example: "To divide (2 + 3i)/(1 − i), we multiply the numerator and denominator by the          (1 + i) to eliminate the imaginary part from the denominator, yielding (−1 + 5i)/2."

Complex Fraction

A complex fraction is a fraction in which the numerator, denominator, or both contain fractions themselves. It is simplified by multiplying both the numerator and denominator by the least common denominator of all inner fractions.

Example: "The          (1/x + 1/y)/(1/x − 1/y) can be simplified by multiplying the numerator and denominator by the LCD xy to get (y + x)/(y − x)."

Complex Number System

The complex number system consists of all numbers of the form a + bi, where a and b are real numbers and i = √(−1). It extends the real number system and guarantees that every polynomial equation has a complete set of roots.

Example: "Within the         , every polynomial of degree n has exactly n roots (counting multiplicity), so the equation x⁴ + 1 = 0, which has no real solutions, still has four complex roots."

Complex Plane

The complex plane (Argand plane) graphs complex numbers using a real x-axis and imaginary y-axis, where x+yi is plotted as point (x,y).

Example: "The complex number 4 − 3i is plotted on the          at coordinates (4, −3), and its distance from the origin—its modulus—is √(16 + 9) = 5."

Complex Root

A complex root is a non-real solution to a polynomial equation of the form a + bi with b ≠ 0. Complex roots of polynomials with real coefficients always appear in conjugate pairs: if a + bi is a root, so is a − bi.

Example: "Since the discriminant of x² + 2x + 5 is 4 − 20 = −16 < 0, the quadratic has no real solutions; each          is −1 ± 2i, appearing as a conjugate pair."

Complex Roots

Complex roots are the non-real solutions to a polynomial equation, always occurring in conjugate pairs for real-coefficient polynomials. By the Fundamental Theorem of Algebra, a degree-n polynomial has exactly n roots in the complex number system.

Example: "The polynomial x⁴ − 1 has four         : 1, −1, i, and −i, and when plotted on the complex plane they are equally spaced on the unit circle at 90° intervals."

Component

A component of a vector is its projection along a coordinate axis. A two-dimensional vector v = ⟨a, b⟩ has horizontal component a and vertical component b, which together fully describe the vector's magnitude and direction.

Example: "The vector v = ⟨−3, 7⟩ has a horizontal          of −3 (pointing in the negative x-direction) and a vertical          of 7 (pointing in the positive y-direction)."

Component Form of a Vector

The component form of a vector expresses it as an ordered pair ⟨a, b⟩ where a is the horizontal displacement and b is the vertical displacement. Given initial point (x₁, y₁) and terminal point (x₂, y₂), the component form is ⟨x₂ − x₁, y₂ − y₁⟩.

Example: "A vector with initial point (2, −1) and terminal point (5, 4) is written in component form as v = ⟨5 − 2, 4 − (−1)⟩ = ⟨3, 5⟩, giving a magnitude of √34."

Compound Inequality

A compound inequality joins two or more inequalities, like 3 < x < 5. Learn what it means, how it works, and see clear examples.

Example: "The          −3 ≤ 2x + 1 < 9 can be solved in parts to yield −2 ≤ x < 4, which is expressed in interval notation as [−2, 4)."

Compound Interest

Compound interest is interest earned on both your original balance and previous interest, making your money grow faster than simple interest.

Example: "An investment of $1000 at an annual rate of 6% compounded monthly grows according to A = 1000(1 + 0.06/12)^(12t), reaching approximately $1819.40 after 10 years."

Compression

Compression in math is a transformation that makes a figure smaller, either toward a point or along a graph axis. Distinct from dilation, which enlarges.

Example: "The function g(x) = f(3x) represents a horizontal          of f by a factor of 1/3, shrinking the graph toward the y-axis so that features appear three times closer together."

Concave Down

Concave down means a curve shaped like an upside-down bowl. Explore the definition, visual examples, and how it differs from concave up.

Example: "On the interval (−∞, 0), the function f(x) = −x⁴ + 2x² is          wherever f''(x) < 0, meaning the curve bends like an upside-down bowl in that region."

Concave Up

Concave up describes a curve shaped like a right-side up bowl. Explore the definition, visual examples, and how it compares to concave down.

Example: "Because f''(x) = 6x and f''(1) = 6 > 0, the function f(x) = x³ is          at x = 1, meaning its curve bends upward like a bowl at that point."

Concavity

Concavity describes the direction in which a curve bends. A function is concave up where it curves like a bowl (opening upward) and concave down where it curves like an inverted bowl. Concavity changes at inflection points.

Example: "The function f(x) = sin(x) alternates          at each inflection point: it is concave down on (0, π) and concave up on (π, 2π), changing sign where f''(x) = −sin(x) = 0."

Conic Section

A conic section is a curve formed by the intersection of a plane with a double cone. The four types are circles, ellipses, parabolas, and hyperbolas, each described by a second-degree equation in x and y.

Example: "The equation 4x² + 9y² = 36 represents an ellipse, one of the four          obtained by slicing a double cone at different angles, the others being the parabola, hyperbola, and circle."

Conjugate

The conjugate of a binomial expression a + b is a − b (and vice versa). Multiplying an expression by its conjugate eliminates radicals or imaginary terms, since (a + b)(a − b) = a² − b².

Example: "To rationalize the denominator of 1/(√5 − 2), we multiply by the          (√5 + 2)/(√5 + 2) to obtain (√5 + 2)/(5 − 4) = √5 + 2."

Conjugate Axis

The conjugate axis of a hyperbola is the segment perpendicular to the transverse axis at the center, with length 2b. It does not intersect the hyperbola but is used to construct the asymptotes and the central rectangle.

Example: "For the hyperbola y²/16 − x²/9 = 1, the          has length 2b = 6 and lies along the x-axis, perpendicular to the transverse axis which runs along the y-axis."

Constant Function

A constant function is a function of the form f(x) = c where c is a fixed real number. Its graph is a horizontal line, its average rate of change over any interval is zero, and it has no x-intercept unless c = 0.

Example: "The function f(x) = 7 is a          whose graph is a horizontal line at y = 7; its average rate of change over any interval is always zero."

Continuity

Continuity at a point x = c requires three conditions: f(c) is defined, the limit of f(x) as x → c exists, and the limit equals f(c). A function continuous on an interval can be graphed without lifting the pen.

Example: "A function has          at x = c if the limit as x approaches c equals f(c); for example, f(x) = x² is continuous everywhere because lim(x→c) x² = c² for all real c."

Continuous

A continuous set is a connected set of numbers like an interval or the real numbers. Unlike discrete sets, no gaps exist between values.

Example: "The exponential function f(x) = eˣ is          on (−∞, ∞), meaning its graph can be traced without lifting a pencil and there are no gaps, holes, or jumps in its behavior."

Continuous at a Point

A function f is continuous at x = c if f(c) is defined, lim(x→c) f(x) exists, and lim(x→c) f(x) = f(c). Failing any one of these three conditions produces a discontinuity at that point.

Example: "The function f(x) = (x² − 4)/(x − 2) is not          x = 2 because f(2) is undefined there, though the limit as x → 2 equals 4."

Continuous Compounding

Continuous compounding is the limiting case of compound interest in which interest is compounded infinitely many times per year. The formula is A = Pe^(rt), where P is the principal, r is the annual rate, and t is time in years.

Example: "Under         , $500 invested at 5% annual interest grows to A = 500e^(0.05 · 8) ≈ $745.91 after 8 years, using the formula A = Pe^(rt)."

Continuous Function

A continuous set is a connected set of numbers like an interval or the real numbers. Unlike discrete sets, no gaps exist between values.

Example: "Because f(x) = ln(x) is a          on (0, ∞), the Intermediate Value Theorem guarantees that for any target value y between ln(1) and ln(10), some x in (1, 10) satisfies f(x) = y."

Contrapositive

Contrapositive: switch and negate hypothesis and conclusion. If P then Q becomes If not Q then not P. Used in logic proofs.

Example: "The conditional 'If f is differentiable at x, then f is continuous at x' has          'If f is not continuous at x, then f is not differentiable at x,' and both statements are logically equivalent."

Convergent Sequence

A convergent sequence has a real number limit. For example, 2.1, 2.01, 2.001… converges to 2. If the limit is infinity, the sequence diverges.

Example: "The sequence aₙ = (2n + 1)/n = 2 + 1/n is a          because as n → ∞, 1/n → 0, so the terms approach the limit 2."

Converse

The converse switches the hypothesis and conclusion of an if-then statement. Example: If it rains, the grass is wet becomes If the grass is wet, it rains.

Example: "The          of 'If f(x) is an even function, then its graph is symmetric about the y-axis' is 'If a graph is symmetric about the y-axis, then f(x) is even,' which is also true."

Coordinate

A coordinate is a number that specifies the position of a point along an axis. In a two-dimensional coordinate system, an ordered pair (x, y) gives the horizontal position x and the vertical position y of a point.

Example: "The          (−2, 5) indicates a point located 2 units left of the y-axis and 5 units above the x-axis on the Cartesian plane, where −2 is the x-         and 5 is the y-        ."

Coordinate Plane

A coordinate plane is a flat surface formed by a horizontal x-axis and vertical y-axis intersecting at the origin. Also called the Cartesian plane.

Example: "When graphing the function f(x) = 2x − 3 on a         , the y-intercept (0, −3) and slope of 2 allow us to plot points and draw the line accurately."

Correlation

Correlation measures how strongly two variables are related. Explore positive and negative associations, with examples like height and weight.

Example: "A scatter plot showing that students who study more hours tend to score higher on exams suggests a positive          between study time and exam score."

Correlation Coefficient

Correlation coefficient measures the strength and direction of the relationship between two variables, expressed as r between -1 and 1.

Example: "A          of r = −0.95 between two variables indicates a strong negative linear relationship, meaning as one variable increases, the other decreases predictably."

Cosecant

Cosecant (csc θ) is the reciprocal of sine in trigonometry. Defined by SOHCAHTOA or the unit circle, it's a periodic function with period 2π.

Example: "For the angle θ = π/6, csc(π/6) = 1/sin(π/6) = 1/(1/2) = 2, confirming that          is the reciprocal of sine and is undefined wherever sin(θ) = 0."