Math 30-1 flashcards
Math 30-1 · Build recall with a focused review session.
32 cards. Read the term and the definition here. The deck at the top of this page is optional.
- Function
- A relation in which each value of the independent variable x is paired with exactly one value of y; it passes the vertical line test.
- Vertical line test
- A graph represents a function only if no vertical line crosses it more than once, because a second crossing would give one x-value two different y-values.
- Transformation mapping
- For y = a·f(b(x − h)) + k, each point (x, y) on y = f(x) moves to (x/b + h, ay + k); h shifts horizontally and k shifts vertically.
- Horizontal line test
- A function has an inverse function only if every horizontal line meets its graph at most once, which means the function is one-to-one.
- Inverse function
- Found by interchanging x and y and solving for y; its graph is the reflection of the original function's graph across the line y = x.
- Composition domain
- The domain of f(g(x)) consists of those x in the domain of g for which g(x) lies in the domain of f.
- Absolute value function
- y = |x| equals x when x ≥ 0 and −x when x < 0; its graph is V-shaped with vertex at the origin and range y ≥ 0.
- Rational function asymptote
- A non-cancelling factor in the denominator produces a vertical asymptote; the horizontal asymptote is found by comparing the degrees of numerator and denominator.
- Radian definition
- One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius; π radians equal 180°.
- Unit circle coordinates
- On the unit circle, the point at angle θ has coordinates (cos θ, sin θ), which immediately gives the identity cos²θ + sin²θ = 1.
- Pythagorean identity
- sin²θ + cos²θ = 1 holds for every angle θ; dividing both sides by cos²θ produces 1 + tan²θ = sec²θ.
- Sinusoidal parameters
- In y = a·sin b(x − c) + d, the amplitude is |a|, the period is 360°/|b| in degrees, the phase shift is c, and the vertical displacement is d.
- Reference angle
- The acute angle between the terminal arm and the x-axis; it gives the magnitude of sine, cosine, and tangent, while the quadrant determines the signs.
- Law of cosines
- c² = a² + b² − 2ab·cos C relates three sides of a triangle to one angle; it applies to SAS and SSS cases.
- Law of sines
- a/sin A = b/sin B = c/sin C applies to triangles; when two sides and a non-included angle are given, there may be zero, one, or two triangles.
- General solution for sine
- If sin θ = k, then θ = arcsin k + 2πn or θ = π − arcsin k + 2πn, where n is any integer.
- Logarithm definition
- log_b x = y means exactly that bʸ = x, where b > 0, b ≠ 1, and x > 0; the logarithm is the exponent required.
- Product law for logarithms
- log_b(MN) = log_b M + log_b N for positive M and N; the logarithm of a product equals the sum of the logarithms of the factors.
- Quotient law for logarithms
- log_b(M/N) = log_b M − log_b N for positive M and N; the logarithm of a quotient equals the difference of the two logarithms.
- Power law for logarithms
- log_b(Mᵖ) = p·log_b M; an exponent inside a logarithm can be moved to the front as a multiplier of the logarithm.
- Change of base formula
- log_b x = log_c x / log_c b, which allows logarithms in any base to be evaluated using base 10 or base e on a calculator.
- Exponential function
- A function of the form y = a·bˣ with b > 0 and b ≠ 1; it shows growth when b > 1, decay when 0 < b < 1, and has y-intercept a.
- Solving exponential equations
- Rewrite both sides with a common base and equate the exponents, or take the logarithm of both sides and then isolate the unknown exponent.
- Half-life model
- A = A₀(1/2)^(t/h), where A₀ is the initial amount, h is the half-life, t is elapsed time, and A is the remaining amount.
- Fundamental counting principle
- If one choice can be made in m ways and a second independent choice in n ways, then the two choices together can be made in m·n ways.
- Factorial
- n! = n(n − 1)(n − 2)···(2)(1) for positive integers n, and 0! is defined to equal 1; n! counts the orderings of n distinct objects.
- Permutation formula
- nPr = n!/(n − r)! counts the ordered arrangements of r objects selected from n distinct objects without replacement.
- Combination formula
- nCr = n!/(r!(n − r)!) counts the unordered selections of r objects from n distinct objects, where a different order of the same objects is not a new selection.
- Symmetry of combinations
- nCr = nC(n − r), because selecting r objects to include is equivalent to selecting n − r objects to exclude from the same group.
- Permutations with identical objects
- The number of distinct arrangements of n objects containing repeated types is n!/(a!b!c!…), where a, b, c are the counts of each identical type.
- Binomial general term
- In the expansion of (a + b)ⁿ, the term containing b^r is nCr·a^(n−r)·b^r, where r takes integer values from 0 to n.
- Sum of binomial coefficients
- The sum of all coefficients in the expansion of (a + b)ⁿ is 2ⁿ, obtained by substituting a = 1 and b = 1.