Bagrut Mathematics (3/4/5 Units) flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a). The expression b² − 4ac is the discriminant.

Difference of squares

The identity a² − b² = (a − b)(a + b). It factors any binomial that is a difference of two perfect squares and is used to simplify rational expressions.

Remainder theorem

When a polynomial P(x) is divided by x − r, the remainder equals P(r). Consequently, x − r is a factor of P(x) exactly when P(r) = 0.

Vieta's formulas

For roots x₁ and x₂ of ax² + bx + c = 0, the sum x₁ + x₂ = −b/a and the product x₁x₂ = c/a. They allow finding root relationships without solving.

Absolute value inequality

For c > 0, |f(x)| < c is equivalent to −c < f(x) < c, while |f(x)| > c is equivalent to f(x) < −c or f(x) > c. For c < 0 the first has no solution.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Adding or subtracting the same quantity from both sides preserves the sign.

Quadratic inequality solution

For a > 0 and roots r₁ < r₂ of ax² + bx + c = 0, ax² + bx + c > 0 holds for x < r₁ or x > r₂, and ax² + bx + c < 0 holds for r₁ < x < r₂.

Extraneous solutions

Solutions introduced by squaring both sides or multiplying by an expression that can be zero. Each candidate must be substituted into the original equation and discarded if it fails.

Even and odd functions

A function is even when f(−x) = f(x) for all x in its domain, giving symmetry about the y-axis. It is odd when f(−x) = −f(x), giving symmetry about the origin.

Inverse function condition

A function has an inverse only if it is one-to-one on its domain. The graph of the inverse is the reflection of the original graph across the line y = x.

Parabola vertex

The graph of y = ax² + bx + c has its vertex at x = −b/(2a). The parabola opens upward when a > 0 and downward when a < 0, with the vertex as maximum or minimum.

Function transformations

Replacing x by x − h shifts the graph h units right; adding k shifts it k units up. Multiplying f(x) by −1 reflects across the x-axis, and replacing x by −x reflects across the y-axis.

Logarithm definition

The equation logₐ b = c means aᶜ = b, valid for a > 0, a ≠ 1, and b > 0. The logarithm is the exponent to which the base must be raised.

Change of base formula

For positive bases a and c different from 1, logₐ b = log꜀ b / log꜀ c. In particular, logₐ b = ln b / ln a, allowing evaluation with natural logarithms.

Exponential function properties

The function f(x) = aˣ has domain all real numbers and range (0, ∞). It passes through (0, 1), is increasing when a > 1, and decreasing when 0 < a < 1.

Solving exponential equations

To solve aˣ = b with b > 0, take logarithms of both sides to obtain x = logₐ b = ln b / ln a. If the bases are equal, equate the exponents directly.

Law of cosines

In a triangle with sides a, b, c and angle C opposite side c, c² = a² + b² − 2ab·cos C. It applies when two sides and the included angle or all three sides are known.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ or sin²θ yields the related identities 1 + tan²θ = 1/cos²θ and 1 + cot²θ = 1/sin²θ.

Triangle area with sine

The area of a triangle with two sides a and b and included angle C is S = ½ab·sin C. This formula is useful when the height is not given directly.

Double angle formulas

sin 2θ = 2 sin θ cos θ, and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. These identities express trigonometric values of twice an angle.

Arithmetic sequence formula

The nth term of an arithmetic sequence is aₙ = a₁ + (n − 1)d, where d is the constant difference between consecutive terms. Its sum is Sₙ = n(a₁ + aₙ)/2.

Geometric series sum

For a geometric series with first term a₁ and ratio q ≠ 1, the sum of the first n terms is Sₙ = a₁(qⁿ − 1)/(q − 1). When |q| < 1, the infinite sum equals a₁/(1 − q).

Infinite geometric series convergence

An infinite geometric series converges only when the absolute value of its ratio q is less than 1, and its sum is a₁/(1 − q). If |q| ≥ 1, the series diverges.

Chain rule

For a composite function, d/dx f(g(x)) = f′(g(x))·g′(x). The derivative of the outer function is evaluated at the inner function and multiplied by the inner derivative.

Critical points

Points in the domain of f where f′(x) = 0 or f′(x) does not exist. They are candidates for local extrema; the sign change of f′ determines whether each is a maximum or minimum.

Tangent line equation

The tangent to y = f(x) at x = a is y − f(a) = f′(a)(x − a). The derivative f′(a) gives the slope of the tangent at that point.

Fundamental theorem of calculus

If F is an antiderivative of f on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a). It links definite integrals with antiderivatives and enables exact area computation.

Power rule for integration

For n ≠ −1, ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C. The special case n = −1 gives ∫x⁻¹ dx = ln|x| + C for x ≠ 0.

Area between curves

If f(x) ≥ g(x) on [a, b], the area between their graphs is ∫ₐᵇ [f(x) − g(x)] dx. The limits a and b are typically intersection x-values of the two curves.

Inscribed angle theorem

An angle inscribed in a circle measures half the intercepted arc. Angles subtending the same arc are equal, and an angle inscribed in a semicircle is a right angle.

Similar triangles

Triangles with equal corresponding angles have proportional corresponding sides. The ratio of their areas equals the square of the ratio of corresponding side lengths.

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1, that is, m₁·m₂ = −1. Parallel lines have equal slopes.

Binomial probability

In n independent trials with success probability p, P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ, where C(n, k) = n!/[k!(n − k)!] counts the arrangements of k successes.